g | x | w | all
Bytes Lang Time Link
019Uiua250929T110245Znoodle p
026Jelly250929T135554ZUnrelate
053Dyalog APL250929T050629ZAaron
104Python 3210705T021125Zaeh5040
nanCommon Lisp210703T132800ZThun_A
055J210703T214616ZJonah
139Excel210703T141107ZAxuary
04105AB1E210703T140024ZMakonede
029Jelly160310T213915ZDennis
124Python 3210702T131737ZJakque
044Vyxal210702T101025Zemanresu
039MATL160310T205330ZLuis Men
127Python 3160310T204636ZSherlock
060Perl160311T125745ZTon Hosp
130Python 3160311T135419ZErwan
131Perl160310T201409ZDoorknob
066Retina160310T194309ZDigital
093JavaScript ES6160310T200851ZETHprodu
178JavaScript ES6160310T205643ZRevanPro
139Lua160311T190427ZEgor Skr
3836Pyth160310T194516ZFryAmThe
125Javascript ES6160310T203414ZCharlie
163Ruby160310T195648ZDoorknob
283Python 3160310T195622ZAdnan
066Pyth160310T192700ZDenker
311JavaScript160310T192013ZJens Ren

Uiua, 19 bytes

⍜(♭⍜⊣⇌⍉⋯⨂" '.:")°↻₁

try it!

Perhaps the only task on this site where Uiua can out-golf Jelly by ten seven bytes. This task is extremely well suited to Uiua because the entire operation can be expressed with a single ⍜ under. For those out of the loop, under takes two functions and applies the first one to some value, then applies the second one, and finally applies an inverse transformation of the first. Here, the first function does the following:

♭⍜⊣⇌⍉⋯⨂" '.:"
      ⨂" '.:"  # zero-indexed position of each in this string
     ⋯         # each converted to binary (little-endian)
    ⍉          # transpose (giving our 2xN array)
 ⍜⊣⇌           # reverse the last row
♭              # flatten the array

From here, all that has to be done is °↻₁ to cycle this flattened form one to the right, and the above transformation is undone, resulting in the desired string.

This is possible because Uiua's implementation of under allows very advanced use-cases like this because each operation stores context which it can use to undo itself in the end. For example, when the array is flattened, Uiua keeps track of the original shape of the array and molds it back while inverting.

Jelly, 26 bytes

“.': ”ðṖiⱮd2j@/ṙ1ŒHUż¥/Ḅịḷ

Try it online!

Granted that is 2016 jelly

Newer answers could be shorter

-- lyxal

Just barely! Jelly changed a lot between 2016 and 2019, but the main byte saver here over Dennis' solution is just not having two separate string constants. There might be a shorter way to perform the core logic as well, but this kind of thing isn't really Jelly's forte--either rearranging multi-dimensional arrays or computing things under some transformation.

        iⱮ                    Find the 1-index of every character in
“.': ” Ṗ                      the string ".': " without the trailing space.
          d2                  Divmod the indices by 2.
              /               Reduce the list of [div, mod] pairs by
            j@                sticking the accumulator between the div and mod.
“.': ” ṖiⱮd2j@/               The result is the 's backwards followed by the .s.
               ṙ1             Rotate that 1 to the left,
                 ŒH           split it in half,
                    ż /       pair the elements of the halves
                   U ¥        with the first half (un-)reversed,
                       Ḅ      convert each pair from binary,
“.': ”ð                 ịḷ    and index into ".': ".

Dyalog APL, 53 bytes

{' ''.:'⌷⍨⊂1+2⊥⊖⌽@2⊢2(≢⍵)⍴¯1⌽,⌽@2⊖2 2⊤⌊7÷⍨32-⍨⎕UCS⍵}­⁡​‎⁠⁠⁠‎⁡⁠⁣⁤⁣‏⁠‎⁡⁠⁣⁤⁤‏⁠‎⁡⁠⁤⁡⁡‏⁠‎⁡⁠⁤⁡⁢‏⁠‎⁡⁠⁤⁡⁣‏⁠‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁣⁣⁣‏⁠‎⁡⁠⁣⁣⁤‏⁠‎⁡⁠⁣⁤⁡‏⁠‎⁡⁠⁣⁤⁢‏⁠‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁣⁢⁣‏⁠‎⁡⁠⁣⁢⁤‏⁠‎⁡⁠⁣⁣⁡‏⁠‎⁡⁠⁣⁣⁢‏⁠‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁣⁡⁣‏⁠‎⁡⁠⁣⁡⁤‏⁠‎⁡⁠⁣⁢⁡‏⁠‎⁡⁠⁣⁢⁢‏⁠‏​⁡⁠⁡‌⁢⁡​‎‎⁡⁠⁣⁡⁢‏⁠‏​⁡⁠⁡‌⁢⁢​‎‎⁡⁠⁢⁤⁣‏⁠‎⁡⁠⁢⁤⁤‏⁠‎⁡⁠⁣⁡⁡‏‏​⁡⁠⁡‌⁢⁣​‎‎⁡⁠⁢⁣⁣‏⁠‎⁡⁠⁢⁣⁤‏⁠‎⁡⁠⁢⁤⁡‏⁠‎⁡⁠⁢⁤⁢‏‏​⁡⁠⁡‌⁢⁤​‎‎⁡⁠⁢⁢⁡‏⁠‎⁡⁠⁢⁢⁢‏⁠‎⁡⁠⁢⁢⁣‏⁠‎⁡⁠⁢⁢⁤‏⁠‎⁡⁠⁢⁣⁡‏⁠‎⁡⁠⁢⁣⁢‏‏​⁡⁠⁡‌⁣⁡​‎‎⁡⁠⁢⁡⁡‏⁠‎⁡⁠⁢⁡⁢‏⁠‎⁡⁠⁢⁡⁣‏⁠‎⁡⁠⁢⁡⁤‏⁠‏​⁡⁠⁡‌⁣⁢​‎‎⁡⁠⁤⁤‏‏​⁡⁠⁡‌⁣⁣​‎‎⁡⁠⁤⁢‏⁠‎⁡⁠⁤⁣‏‏​⁡⁠⁡‌⁣⁤​‎‎⁡⁠⁣⁤‏⁠‎⁡⁠⁤⁡‏‏​⁡⁠⁡‌⁤⁡​‎‎⁡⁠⁢‏⁠‎⁡⁠⁣‏⁠‎⁡⁠⁤‏⁠‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏⁠‎⁡⁠⁢⁣‏⁠‎⁡⁠⁢⁤‏⁠‎⁡⁠⁣⁡‏⁠‎⁡⁠⁣⁢‏⁠‎⁡⁠⁣⁣‏‏​⁡⁠⁡‌­
                                              ⎕UCS⍵   # ‎⁡Get codepoint
                                          32-⍨        # ‎⁢subtact 32 (space (32) being the smallest of the 4 chars)
                                      ⌊7÷⍨            # ‎⁣divide by 7 and floor
                                  2 2⊤                # ‎⁤Encode each of those as binary (2 digits)
                                 ⊖                    # ‎⁢⁡Flip it over
                              ⌽@2                     # ‎⁢⁢Replace 2nd row with itself reversed
                          ¯1⌽,                        # ‎⁢⁣Ravel and shift to the right
                                                      # ‎⁢⁣This gives the conveyor belt shift
                    2(≢⍵)⍴                            # ‎⁢⁤Reshape that to 2xN (the original input length)
                ⌽@2⊢                                  # ‎⁣⁡Re-reverse 2nd row again
               ⊖                                      # ‎⁣⁢And flip back over
             2⊥                                       # ‎⁣⁣Decode the binary number
           1+                                         # ‎⁣⁤Add 1 for indexing's sake
 ' ''.:'⌷⍨⊂                                           # ‎⁤⁡And finally select from magic string
💎

Created with the help of Luminespire.

Python 3, 105 104 bytes

s=" .':"
a=[*map(s.find,input())]
b=a[0]*2,*a,a[-1]//2
print(*map(lambda x,y:s[x&2|y&1],b,b[2:]),sep='')

Try it online!

Common Lisp, 253 247 bytes

(defun R(i &aux(n(length i))(u #1=(make-list n))(b #1#))(dotimes(k n)(setf(elt u k)(find #2=(elt i k)"':")(elt b k)(find #2#".:")))(map'string(lambda(u b)(cond((and u b)#\:)(u #\')(b #\.)(t #\ )))`(,(car b),@(butlast u))`(,@(cdr b),(nth(1- n)u))))

Try it online!

Tests included in TIO.

(defun R (i &aux (n (length i))  ; take the length of input
                 (u #1=(make-list n))   ; create the top (up) list
                 (b #1#))  ; create the bottom list
  (dotimes (k.. n)
    (setf (elt u k) (find #2=(elt i k) "':") 
; set the k'th element of the top list to a non-falsey value if k'th element of input is #\' or #\:
          (elt b k) (find #2# ".:")
; set the k'th element of the bottom list to a non-falsey value if k'th element of input is #\. or #\:
(map 'string  ; collect the results into a string
     (lambda (u b) ; for each element of the rotated up and bottom lists
       (cond
        ((and u b) #\:) ; if both top and bottom are non-falsey add #\:
        (u #\')         ; if only top is non-falsey add #\'
        (b #\.)         ; if only bottom is non-falsey add #\.
        (t #\ )))       ; if both are falsey add '#\ ' (#\Space)
     `(,(car b),@(butlast u))   ; first element of the bottom and the all but last element of the top
     `(,(cdr b),(nth(1- n)u)))) ; all but first element of the bottom and the last element of the top

Saved 6 bytes using #= notation for (make-list n) and (elt i k)

J, 55 bytes

' .'':'&([{~([:g@($$_1|.,)g=.|.@{:1}_2{.])&.(|:@#:)@i.)

Try it online!

Excel, 139 bytes

=LET(e,LEN(A1),q,SEQUENCE(e),x,CODE(MID(A1,q,1))-32,l,x>7,u,x>l*14,CONCAT(MID(" '.:",IF(q=1,l,INDEX(u,q-1))+IF(q=e,u,INDEX(l,q+1))*2+1,1)))

Link to Spreadsheet

How it works

=LET(
*Assignments*
e,LEN(A1),              'e = length of A1
q,SEQUENCE(e),          'q = (1 .. e)
x,CODE(MID(A1,q,1))-32, 'x = list of ASCII codes of each character - 32
l,x>7,                  'l = array of lower numbers = 1 if x>7 (.:)
u,x>l*14,               'u = array of upper numbers = 1 if x>l*14 (':)

*Result*
IF(q=1,l,INDEX(u,q-1)) +   'String indices = if q = 1 then l[1] else u[q-1] 
IF(q=e,u,INDEX(l,q+1))*2+1 '               + (if q=e then u[e] else l[q+1]) * 2 + 1
CONCAT(MID(" '.:",~,1)))   'Concatenate all the results

05AB1E, 41 bytes

Â)εS„':„.:‚Nès夈¨}R¯øí˜2äćR¸ìζJC" '.:"sè

Try it online! Outputs as a list of characters. Link includes a footer to format the output.

Â)εS„':„.:‚Nès夈¨}R¯øí˜2äćR¸ìζJC"..."sè  # trimmed program
                                       è  # push character...
                                          # (implicit) s...
                                       è  # in...
                                 "..."s   # literal...
                                       è  # at...
                                      s   # (implicit) indices in...
                              ζ           # list of elements with same indices in each element of...
                            ¸             # list of...
                          ć               # first element of...
 )                                        # list of...
                                         # implicit input...
 )                                        # and...
                                          # implicit input...
                                         # reversed...
  ε                                       # with each element replaced by...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                 ¨                        # excluding the last element...
                ˆ                         # and append...
               ¤                          # last element of...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                ˆ                         # to global array...
                   R                      # reversed...
                      í                   # with...
                    ¯                     # global array...
                     ø                    # with each element paired with corresponding element in...
                    ¯                     # global array...
                      í                   # prepended...
                       ˜                  # flattened...
                         ä                # split into...
                        2                 # literal...
                         ä                # equal pieces...
                           R              # reversed...
                             ì            # with...
 )                                        # list of...
                                         # implicit input...
 )                                        # and...
                                          # implicit input...
                                         # reversed...
  ε                                       # with each element replaced by...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                 ¨                        # excluding the last element...
                ˆ                         # and append...
               ¤                          # last element of...
              å¤                          # is...
             s                            # (implicit) each element of...
   S                                      # list of characters of...
                                          # (implicit) current element in map...
              å                           # in...
            ès                            # element in...
          ‚                               # list of...
    „':                                   # literal...
          ‚                               # and...
       „.:                                # literal...
            è                             # at index...
           N                              # current index in map...
                ˆ                         # to global array...
                   R                      # reversed...
                      í                   # with...
                    ¯                     # global array...
                     ø                    # with each element paired with corresponding element in...
                    ¯                     # global array...
                      í                   # prepended...
                       ˜                  # flattened...
                         ä                # split into...
                        2                 # literal...
                         ä                # equal pieces...
                          ć               # excluding the first element...
                             ì            # prepended...
                                          # (implicit) with each element...
                               J          # joined...
                                          # (implicit) with each element...
                                C         # converted from binary to decimal

Jelly, 32 30 29 bytes

,Ṛe€"“':“.:”µṪ€;"ṚU1¦ZḄị“'.: 

Note the trailing space. Try it online! or verify all test cases.

Background

We begin by considering the input string (e.g., :..:'.) and its reverse.

:..:'.
.':..:

For each character in the top row, we check if it belongs to ':, and for each character of the bottom row if it belongs to .:. This gives the 2D array of Booleans

100110
101111

which is the matrix from the question, with reversed bottom row.

We remove the last Boolean of each row, reverse the order of the rows, prepend the Booleans in their original order, and finally reverse the top row.

100110    10011    10111    010111    111010
101111    10111    10011    110011    110011

This yields the rotated matrix from the question.

Finally, we consider each column of Booleans a binary number and index into '.: to obtain the appropriate characters.

332031    ::. :'

How it works

,Ṛe€"“':“.:”µṪ€;"ṚU1¦ZḄ‘ị“'.:   Main link. Argument: S (string)

 Ṛ                              Reverse S.
,                               Form a pair of S and S reversed.
     “':“.:”                    Yield ["':" ".:"].
  e€"                           For each character in S / S reversed, check if it
                                is an element of "':" / ".:".
                                Yield the corresponding 2D array of Booleans.

            µ                   Begin a new, monadic chain.
                                Argument: A (2D array of Booleans)
             Ṫ€                 Pop the last Boolean of each list.
                 Ṛ              Yield the reversed array of popped list.
               ;"               Prepend the popped items to the popped lists.
                  U1¦           Reverse the first list.
                     Z          Zip to turn top and bottom rows into pairs.
                      Ḅ         Convert each pair from base 2 to integer.
                        “'.:    Yield "'.: ".
                       ị        Retrieve the characters at the corr. indices.

Python 3, 124 bytes

def f(a,s=" .':",j="".join):q=j(bin(4+s.find(i))for i in a);return j(s[int(j(x),2)]for x in zip(q[4]+q[3::5],q[9::5]+q[-2]))

Try it online!

How it works :

If a is equal to

ABCD
EFGH

then q is equal to ---ae---bf---cg---dh with a to h equal to "1" if there is a dot in the corresponding emplacement and "0" otherwise (and --- equal to 0b1)

EABC
FGHD

Vyxal, 44 bytes

f`:'. `f⁺↑b2ẇĿ2ẇÞT÷ḣ„ṫ$„p‟JWÞT⁺↑b2ẇ`:'. `fĿ∑

Try it Online!

Yuck.

MATL, 40 39 bytes

' ''.:'tjw4#mqBGnXKq:QKEh1Kq:K+hv!)XBQ)

Try it online! The linked version has v repaced by &v, because of changes in the language after this answer was posted.

' ''.:'               % pattern string. Will indexed into, twice: first for reading 
                      % the input and then for generating the ouput
t                     % duplicate this string
j                     % input string
w                     % swap
4#m                   % index of ocurrences of input chars in the pattern string
qB                    % subtract 1 and convert to binay. Gives 2-row logical array
GnXKq:QKEh1Kq:K+hv!   % (painfully) build two-column index for rotation
)                     % index into logical array to perform the rotation
XBQ                   % transform each row into 1, 2, 3 or 4
)                     % index into patter string. Implicitly display

Python 3, 166 154 153 150 146 138 137 135 132 127 bytes

Edit: I've borrowed the use of zip from Erwan's Python answer at the end of the function. and their idea to use [::-1] reversals, though I put in my own twist. As it turns out, reversals were not a good idea for my function. I have changed my use of format for further golfing. Moved a and b directly into zip for further golfing (ungolfing remains unchanged because the separation of a and b there is useful for in avoiding clutter in my explanation)

Edit: Borrowed (some number)>>(n)&(2**something-1) from this answer by xnor on the Music Interval Solver challenge. The clutter that is zip(*[divmod(et cetera, 2) for i in input()]) can probably be golfed better, though I do like the expediency it grants from using two tuples t and v.

t,v=zip(*[divmod(708>>2*(ord(i)%5)&3,2)for i in input()])
print("".join(" '.:"[i+j*2]for i,j in zip((v[0],*t),(*v[1:],t[-1]))))

Ungolfed:

def rotate_dots(s):
    # dots to 2 by len(s) matrix of 0s and 1s (but transposed)
    t = []
    v = []
    for i in s:
        m = divmod(708 >> 2*(ord(i)%5) & 3, 2)
            # ord(i)%5 of each char in . :' is in range(1,5)
            # so 708>>2 * ord & 3 puts all length-2 01-strings as a number in range(0,4)
            # e.g. ord(":") % 5 == 58 % 5 == 3
            # 708 >> 2*3 & 3 == 0b1011000100 >> 6 & 3 == 0b1011 == 11
            # divmod(11 & 3, 2) == divmod(3, 2) == (1, 1)
            # so, ":" -> (1, 1)
        t.append(m[0])
        v.append(m[1])

    # transposing the matrix and doing the rotations
    a = (v[0], *t)          # a tuple of the first char of the second row 
                            # and every char of the first row except the last char
    b = (v[1:], t[-1])      # and a tuple of every char of the second row except the first
                            # and the last char of the first row

    # matrix to dots
    z = ""
    for i, j in zip(a, b):
        z += " '.:"[i + j*2]    # since the dots are binary
                                # we take " '.:"[their binary value]
    return z

Perl, 70 69 64 63 61 60 bytes

Includes +2 for -lp

Run with the input string on STDIN, e.g.

perl -lp rotatedots.pl <<< ":..:'."

rotatedots.pl:

y/'.:/02/r=~/./;y/.':/01/;$_=$'.2*chop|$&/2 .$_;y;0-3; '.:

Explanation

y/'.:/02/r                                        Construct bottom row but
                                                  with 2's instead of 1's
                                                  Return constructed value
                                                  (for now assume space
                                                  becomes 0 too)
          =~/./                                   Match first digit on bottom
                                                  row into $&. $' contains
                                                  the rest of the bottom row
                y/.':/01/                         Convert $_ to top row
                                                  (again assume space
                                                  becomes 0 too)
                             $'.2*chop            Remove last digit from
                                                  the top row, multiply by 2
                                                  and append to bottom row
                                       $&/2 .$_   Divide removed digit by
                                                  2 and prepend it to the
                                                  top row
                          $_=         |           "or" the top and bottom
                                                  row together. The ASCII
                                                  values of 0,1,2,3 have
                                                  00,01,10,11 as their last
                                                  two bits.

y;0-3; '.:                  Convert the smashed together top and bottom rows
                            to the corresponding representation characters.
                            Drop the final ; since it is provided by -p
                            (after a newline which doesn't matter here)

Space is not converted in the above code. For the calculations /2 and *2 it will behave like and become 0. In the other positions it will be part of the "or", but the 1 bits of space are a subset of the one bits of 0 and will have the same effect as 0 if or-ed with any of the digits. Only if the character it is or-ed with is a space will it remain a space instead of becoming a 0. But that's ok since 0 would have been converted back to space anyways.

Python 3, 145 141 130 bytes

def f(s):a=[i in"':"for i in s]+[i in".:"for i in s][::-1];return''.join(" '.:"[i+2*j]for i,j in zip([a[-1]]+a,a[-2:len(s)-2:-1]))

Explanation

The golfed solution use the following property of zip : zip('ABCD', 'xy') --> Ax By so zip(a[:l],a[l:]) can be replace by zip(a,a[l:]) and that allow to remove the definition of l

def f(s):
 l=len(s)-1
 #                ┌───── unfold input string :  123  -> 123456
 #                │                             654
 #  ──────────────┴──────────────────────────────
 a=[i in"':"for i in s]+[i in".:"for i in s][::-1]
 # ─────────┬─────────   ────────────┬───────────
 #          │                        └──── generate the second row and reverse it
 #          └─────────── generate the first row 

 return''.join(" '.:"[i+2*j]for i,j in zip([a[-1]]+a[:l],a[l:-1][::-1]))
 #             ──────┬──────           ─┬    ────────────┬───────────
 #                   │                  │                └──── rotate and create first/second new row :  123456  -> 612345  -> 612
 #                   │                  │                                                                                      543
 #                   │                  └ group pair of the first and second row : 612 -> (6,5),(1,4),(2,3)
 #                   │                                                             543
 #                   └─────────── replace pair by symbol 

Results

>>> f(".'  '.::  :.'. . ::.'  '. . .::'  :.'.")
"' ' .:.''..'.'. ..:' ' .'. ...'''..'.'"
>>> f(".....''''''")
":...  '''':"

Perl, 144 142 137 131 bytes

y/.':/1-3/;s/./sprintf'%02b ',$&/ge;@a=/\b\d/g;@b=(/\d\b/g,pop@a);@a=(shift@b,@a);say map{substr" .':",oct"0b$a[$_]$b[$_]",1}0..@a

Byte added for the -n flag.

Pretty much the same algorithm as my Ruby answer, just shorter, because... Perl.

y/.':/1-3/;                         # transliterate [ .':] to [0123]
s/./sprintf'%02b ',$&/ge;           # convert each digit to 2-digit binary
@a=/\b\d/g;                         # grab the 1st digit of each pair
@b=(/\d\b/g,                        # 2nd digit of each pair
pop@a);                             # push the last element of a to b
@a=(shift@b,@a);                    # unshift the first element of b to a
say                                 # output...
map{                                # map over indices of a/b
substr" .':",oct"0b$a[$_]$b[$_]",1  # convert back from binary, find right char
}0..@a                              # @a is length of a

Obnoxiously, @a=(shift@b,@a) is shorter than unshift@a,shift@b.

Alas, these are the same length:

y/ .':/0-3/;s/./sprintf'%02b ',$&/ge;
s/./sprintf'%02b ',index" .':",$&/ge;

Thanks to Ton Hospel for 5 bytes and msh210 for a byte!

Retina, 66

:
1e
\.
1f
'
0e
 
0f
T`h`Rh`^.|.$
(.)(\d)
$2$1
e1
:
e0
'
f0
 
f1
.

Explanation

Starting with input:

: ''. :

The first 4 stages construct the matrix, using 1/e for true and 0/f for false for the top/bottom rows, respectively. Top and bottom rows are interlaced together. This would yield a string like:

e1f0e0e0f1f0e1

However, these 4 stages also effectively move the lower row 1 to the left, simply by reversing the order of the letters and digits:

1e0f0e0e1f0f1e

The Transliteration stage reverses hex digits for the first and last characters only, i.e. replaces 0-9a-f with f-a9-0. This has the effect of moving the bottom-left character up to the top row and the top-right character down to the bottom row:

ee0f0e0e1f0f11

The next stage then swaps every letter-digit pair, thereby moving the upper row 1 to the right. Previously this was (\D)(\d), but it turns out that (.)(\d) is sufficient because the substitutions always happen left-to-right and so the final two digits won't be erroneously matched by this, because penultimate character will have been already substituted. The matrix has now been fully rotated as required:

e0e0f0e1e0f1f1

The final 4 stages then translate back to the original format:

'' :'..

Try it online.

All testcases, one per line, m added to T line to allow separate treatment of each input line.

JavaScript (ES6), 100 97 93 bytes

Saved 4 bytes thanks to @edc65

s=>s.replace(/./g,(c,i)=>" '.:"[(i?q>' '&q!=".":c>"'")+(q=c,s[++i]?s[i]>"'":c>' '&c!=".")*2])

How it works

This decides on the character we need to insert by performing some calculations on the chars before and after the current one. We sum:

This sums nicely to 0 for a space, 1 for ', 2 for ., and 3 for :.

Test snippet

f=s=>s.replace(/./g,(c,i)=>" '.:"[(i?q>' '&q!=".":c>"'")+(q=c,s[++i]?s[i]>"'":c>' '&c!=".")*2])
Try it out:
<input id=A type="text" value=".'  '.::  :.'. . ::.'  '. . .::'  :.'.">
<button onclick="B.innerHTML=f(A.value)">Run</button>
<pre id=B></pre>

JavaScript (ES6), 237 210 204 188 182 178 bytes

Credit to @Downgoat for saving 16 bytes in the 188-byte revision

Update: I had a brainwave and reduced the first operation on s to a single map call instead of two separate ones

s=>(r=" .':",a=[],s=[...s].map(c=>(t=('00'+r.indexOf(c).toString(2)).slice(-2),a.push(t[0]),t[1])),a.splice(0,0,s.shift()),s.push(a.pop()),a.map((v,i)=>r[+('0b'+v+s[i])]).join``)

Pretty Print & Explanation

s => (
  r = " .':", // Map of characters to their (numerical) binary representations (e.g. r[0b10] = "'")
  a = [],     // extra array needed
  // Spread `s` into an array
  s = [...s].map(c => (
    // Map each character to a `0`-padded string representation of a binary number, storing in `t`
    t = ('00' + r.indexOf(c).toString(2)).slice(-2)),
    // Put the first character of `t` into `a`
    a.push(t[0]),
    // Keep the second character for `s`
    t[1]
  )),
  // Put the first character of `s` in the first index of `a`
  a.splice(0,0,s.shift()),
  // Append the last character of `a` to `s`
  s.push(a.pop(),
  // Rejoin the characters, alternating from `a` to `s`, representing the rotated matrix, and map them back to their string representation
  // Use implicit conversion of a binary number string using +'0b<num>'
  a.map((v,i) => r[+('0b' + v + s[i])]).join``
)

Lua, 139 bytes

print(((...):gsub(".",{[" "]="NN@",["."]="YN@",["'"]="NY@",[":"]="YY@"}):gsub("(.)@(.?)","%2%1"):gsub("..",{NN=" ",NY=".",YN="'",YY=":"})))

Usage:

$ lua conveyor.lua ".'  '.::  :.'. . ::.'  '. . .::'  :.'."
' ' .:.''..'.'. ..:' ' .'. ...'''..'.'

Pyth, 38 36

L,hb_ebsXCyc2.>syCXzJ" .':"K.DR2T1KJ

2 bytes thanks to Jakube!

Try it here or run the Test Suite.

Explanation:

L,hb_eb         ##  Redefine the function y to take two lists
                ##  and return them but with the second one reversed
                ##  Uses W to apply a function only if it's first argument is truthy
XzJ" .':"K.DR2T ##  Does a translation from the string " .':" to
                ##  .DR2T which is [0,1,2,3...,9] mapped to divmod by 2
                ##  (which is [0,0],[0,1],[1,0],[1,1], then some extra, unused values)
                ##  we also store the string and the list for later use in J and K
.>syC ... 1     ##  zip the lists to get the bits on top and below as two separate lists
                ##  apply the function y from before, flatten and rotate right by 1
Cyc2            ##  split the list into 2 equal parts again, then apply y and zip again
sX ... KJ       ##  apply the list to string transformation from above but in reverse
                ##  then flatten into a string

Javascript ES6 125 bytes

q=>(n=[...q].map(a=>(S=` .':`).indexOf(a))).map((a,i)=>(i?n[i-1]&2:n[0]&1&&2)|((I=n[i+1])>-1?I&1:n[i]&2&&1)).map(a=>S[a]).join``

I map each character to a two digit binary equivalent

 : becomes 3   11
 ' becomes 2   10
 . becomes 1   01
   becomes 0   00

and I'm thinking of them as being one on top of the other

3212021 becomes
1101010
1010001

I save that to n

For each character (0-3) of n, I check it's neighbors, adding the highest order bit of the left neighbor to the lowest order bit of the right neighbor. if i==0 (first character) I use it's own lower order bit instead of the left neighbor's higher order bit.

if n[i+1]>-1 it means we got 0,1,2,3 so when that's false we hit the last element.

When that happens I use the character's own highest order bit instead of the right neighbor's lower bit

map that back to .': land and join that array back together

Ruby, 166 163 bytes

->s{a=s.tr(f=" .':",t='0-3').chars.map{|x|sprintf('%02b',x).chars}.transpose;a[1]+=[a[0].pop];a[0]=[a[1].shift]+a[0];a.transpose.map{|x|x.join.to_i 2}.join.tr t,f}

Yuck... transpose is too long.

Tricks used here:

Python 3, 294 287 283 bytes

Waaayyyyyy too long, but I will try to golf of some bytes:

z=input()
x=len(z)
M=[0,1,2,3]
for Q in M:z=z.replace(":'. "[Q],"11100100"[Q*2:Q*2+2])
a=[]
b=[]
for X in range(x):a+=[z[X*2]];b+=[z[X*2+1]]
b=b[1:]+[a.pop()]
c=[b[0]]+a
z=""
for X in range(len(c)):
 y=c[X]+b[X]
 for Q in M:y=y.replace("11100100"[Q*2:Q*2+2],":'. "[Q])
 z+=y
print(z)

Pyth, 66 bytes

KlQJ.nCm@[,1Z,Z1,ZZ,1 1)%Cd5Qjkm@" .':"id2Ccs[:JKhK<JtK>JhK:JtKK)K

Try it here!

Explanation

This can be broken down in 3 parts:

Convert input

This is fairly trivial. Each character is mapped in the following way:

  -> (0,0)
. -> (0,1)
' -> (1,0)
: -> (1,0)

First one is a whitespace.
We get a list of 2-tuples which we transpose to get the 2 rows of the matrix which then gets flattened.

Code

KlQJ.nCm@[,1Z,Z1,ZZ,1 1)%Cd5Q   # Q = input

KlQ                             # save the width of the matrix in K (gets used later)
       m                    Q   # map each character d
                        %Cd5    # ASCII-code of d modulo 5
        @[,1Z,Z1,ZZ,1 1)        # use this as index into a lookup list
   J.nC                         # transpose, flatten and assign to J
                        

Rotate

We have the matrix as flat array in J and the width of the matrix in K. The rotation can be described as:

J[K] + J[:K-1] + J[K+1:] + J[K-1]

Code

s[:JKhKJhK:JtKK)    # J = flat array, K = width of matrix

s[                  )    # Concat all results in this list
  :JKhK                  # J[K]
       JhK          # J[K+1:]
               :JtKK     # J[K-1]

Convert it back

jkm@" .':"id2Cc[)K    # [) = resulting list of the step above

              c[)K    # chop into 2 rows
             C        # transpose to get the 2-tuples back
  m                   # map each 2-tuple d
          id2         # interpret d as binary and convert to decimal
   @" .':"            # use that as index into a lookup string to get the correct char
jk                    # join into one string

JavaScript, 311 bytes

Can probably be improved alot:

a=(s=prompt()).length-1;o=s[0]==":"||s[0]=="."?s[1]==":"||s[1]=="."?":":"'":s[1]==":"||s[1]=="."?".":" ";for(i=1;i<a;i++)o+=s[i-1]=="'"||s[i-1]==":"?s[i+1]=="."||s[i+1]==":"?":":"'":s[i+1]=="."||s[i+1]==":"?".":" ";alert(o+=s[a]==":"||s[a]=="'"?s[a-1]==":"||s[a-1]=="'"?":":"'":s[a-1]==":"||s[a-1]=="'"?".":" ")