Ruby, 89 bytes

a=[*?a..?z].join;v=[];u=1..-1
27.times{v<<(a=a[u]+a[0])+a.reverse[u]}
puts v+v.reverse[u]

Try it online!

This solution is loosely based on other answers, for instance, this one.

Japt -R, 15 12 bytes

;26ÆCé´T êÃê

Test it

;26ÆCé´T êÃê
 26Æ             :Map the range [0,26)
;   C            :  Lowercase alphabet
     é           :  Rotate right by
      ´T         :    T (initially 0) prefix decremented
         ê       :  Palindromise
          Ã      :End map
           ê     :Palindromise
                 :Implicit output joined with newlines

Elixir, 111 bytes

import Enum
c=?a..?z
a=map 1..26,&(drop(c,&1)++slice c,0,&1)
map a++[hd a]++reverse(a),&IO.puts&1++tl reverse&1

Try it online!

Feel like there are some key savings I'm missing here...

Vyxal jr, 11 bytes

27ƛkaǓøm;øm

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Explanation:

27ƛ     ;    # For each number x in range [1..27]:
   ka        #   Lowercase alphabet
     Ǔ       #   Rotate it left x times
      øm     #   Mirror without duplicating middle
         øm  # Mirror without duplicating middle
             # 'j' flag - Join top of stack with newlines and print

FALSE, 137 bytes

1c:53 0$d:[65@$0=~\1-\][@@\c;+d;1=[1_c:]?$26=[%0 1d:]?$1_=[%25]?$@+1b:51[$0=~][$26=[1_b:]?\$,b;+[$a:$64=[90a:]?91=[65a:]?a;]!\1-]#%%10,]#

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Java 10, 154 153 151 148 147 142 141 bytes

v->{var r="";for(int i=0,j,d=1;i<53;d+=1-++i/27*2){for(j=d;j<26+d;)r+=(char)(j++%26+97);for(j--;j>d;)r+=(char)(--j%26+97);r+="\n";}return r;}

Port of @HatsuPointerKun C++ answer, so make sure to upvote him/her as well.
-8 bytes thanks to @ceilingcat.
-1 byte thanks to @Grimmy.

Try it here.

C++, 191 179 166 165 139 bytes

-12 bytes thanks to Kevin Cruijssen
-14 bytes thanks to Zacharý -26 bytes thanks to ceilingcat

#import<cstdio>
#define C j;)putchar(97+j
main(){for(int i=0,j,d=1;i<53;d+=i++/26?-1:1){for(j=d;26+d>C++%26);for(j--;d<=--C%26);puts("");}}

Stax, 13 bytes

éÑ8╟╘"╧ö↑D☼▓}

Run and debug it

PowerShell, 64 bytes

1..27+26..1|%{$y=$_
-join(0..25+24..0|%{[char](97+($_+$y)%26)})}

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PowerShell v6+, 59 bytes

1..27+26..1|%{-join(('a'..'z'*3)[$_..($_+25)+($_+24)..$_])}

Try it online!

Ranges on ranges on ranges.

05AB1E, 9 bytes

Aā._Ć€ûû»

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APL (Dyalog Classic), 21 bytes

(⍉⊢⍪1↓⊖)⍣2⍉1↓27 27⍴⎕a

Try it online!

With the alphabet (⎕a) fill in row-major order a 27x27 matrix (27 27⍴), then drop the first row (1↓) and transpose (⍉).

Do this twice (⍣2): concatenate to the bottom (⍪) of this matrix (⊢) a vertically reversed copy (⊖) without its first row (1↓) and transpose (⍉) the result.

Yabasic, 74 bytes

Now I'm telling you that this challenge BASIC. /sarcasm

For i=-26To 26
For j=-25To 25
?Chr$(65+Mod(Abs(j)+Abs(i),26));
Next
?
Next

Try it online!

Visual Basic .NET (Mono), 160 bytes

Declared Subroutine that takes no input and outputs to the console.

Module M
Sub Main
Dim S,I,J
For I=-26To 26
S=""
For J=-25To 25
S+=Chr(65+(52-(Math.Abs(j)+Math.Abs(i)))Mod 26)
Next
Console.WriteLine(S)
Next
End Sub
End Module

Try it online!

MY-BASIC, 89 bytes

Anonymous function that takes no input and outputs to the console.

For i=-26 To 26
For j=-25 To 25
Print Chr(65+(52-(Abs(j)+Abs(i)))Mod 26)
Next
Print;
Next

Try it online!

uBASIC, 86 bytes

0ForI=-26To26:ForJ=-25To25:?Left$(Chr$(65+(52-(Abs(J)+Abs(I)))Mod26),1);:NextJ:?:NextI

Try it online!

VBA, _{109 105} 78 Bytes

Anonymous VBE immediate window function that takes no input and outputs the alphabet diamond to the VBE immediate window.

For i=-26To 26:For j=-25To 25:?Chr(65+(52-(Abs(j)+Abs(i)))Mod 26);:Next:?:Next

VBA (Excel) , 116 Bytes

Sub a()
For i=-26To 26
For j=-25To 25
b=b & Chr(65+(52-(Abs(j)+Abs(i))) Mod 26)
Next
Debug.Print b
b=""
Next
End Sub

Following Sir Joffan's Logic. :D

Recursiva, 44 25 21 19 bytes

{p+)'a'"Pp+J''_Q)}}

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Explanation:

{p+)'a'"Pp+J''_Q)}}
{                     - For each
 p+)'a'               - 'abc..xyzazyx..abc'
       "              - Iteration command begin
        P             - Print
         p            - palindromize 'cde..xyzabazyx..cde'
          +           - concatenate 'cde..xyzab'
           J''        - join with nothing 'cde..xyza'
              _       - reverse ['cde...xyz', 'a']
               Q)}    - split ) at } ['a','cdef..xyz']
                  }   - current element (say 'b' for 2nd iteration)

Retina, 166 bytes

bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb____________________________________________________
+(`([a-z]+)_(_+)
$1¶$1$2¶$1
(.)(.{24})(.)(.{24})._
$2$3$1$3$4

Try it online!

Kotlin, 106 bytes

{(0..52).map{i->(0..50).map{j->print((90-((Math.abs(j-25)+Math.abs(i-26)-1)+26)%26).toChar())}
println()}}

Beautified

{
    (0..52).map {i->
        (0..50).map {j->
            print((90 - ((Math.abs(j - 25) + Math.abs(i - 26) - 1)+26) % 26).toChar())
        }
        println()
    }
}

Test

var v:()->Unit =
{(0..52).map{i->(0..50).map{j->print((90-((Math.abs(j-25)+Math.abs(i-26)-1)+26)%26).toChar())}
println()}}

fun main(args: Array<String>) {
    v()
}

TryItOnline

Port of @Karl Napf's answer

Charcoal, 24 21 bytes

-3 bytes thanks to ASCII-only.

Ｆ²⁷«Ｐ✂×β³⊕ι⁺ι²⁷↓»‖Ｂ↓→

Try it online! Link is to verbose version.

...I need to work on my Charcoal-fu. :P

q/kdb+, 51 bytes

Solution:

-1 .Q.a t{rotate[y;x]x,1_(|)x}/:(1+t),1,(-)t:(!)26;

Explanation:

Build the range of 1..26,1,0,-1..-25, feed this into a lambda that rotates the range 0..25 and indexes in at the range 0..25,24..0... Then use this to index into the lowercase alphabet.

-1 .Q.a t{rotate[y;x]x,1_reverse x}/:(1+t),1,neg t:til 26; / ungolfed
-1                                                       ; / print to stdout and swallow return value
                                                 t:til 26  / store range of 0..25 in variable t
                                             neg           / negate (0..-25)
                                           1,              / prepend 1
                                     (1+t),                / prepend 1+t (1..26)
        t{                        }/:                      / lambda function with parameters t, and each-right
                         reverse x                         / x = t = 0..25, reversed is 25..0
                       1_                                  / 1 drop, so 24..0
                     x,                                    / prepend x, so 0..25,24..0
          rotate[y;x]                                      / rotate x by y (rotate[1;"abc"] -> "bca")
   .Q.a                                                    / index into alphabet at these indices

Notes:

Feels like it should be able to be golfed further, building the ranges seems clunky.

Dyalog APL, 31 bytes

{⍵⍪1 0↓⊖⍵}{⍵,0 1↓⌽⍵}↑(⌽∘⎕A)¨⍳27

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Common Lisp, 221 bytes

(let((k 26)a(s"~26{~a~}~25{~a~}~%"))(#4=dotimes(i k)(push(code-char(- 90 i))a))(let*((u(append a a a))(b(reverse u)))(#4#(i 27)(#3=format t s #1=(nthcdr(1+ i)u)#2=(nthcdr(- k i)b)))(rotatef u b)(#4#(i k)(#3#t s #2##1#))))

Try it online!

More readable version:

(let (a)                             ; variable to hold the alphabet
  (dotimes (i 26)                    ; creation of the alphabet as list
    (push (code-char (- 90 i)) a))
  (let* ((u (append a a a))          ; make three copies of a
         (b (reverse u)))                ; reverse u
    (dotimes (i 27)                  ; first half: form a line with two parts
       (format t "~26{~a~}~25{~a~}~%" (nthcdr (1+ i) u) (nthcdr (- 26 i) b)))
    (rotatef u b)                    ; exchange the role of u and b
    (dotimes (i 26)                  ; second half: exchange the two parts
       (format t "~26{~a~}~25{~a~}~%" (nthcdr (- 26 i) b) (nthcdr (1+ i) u)))))

SOGL V0.12, 10 bytes

zl{«:}«¹╬,

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Explanation:

z           push the lowercase alphabet
 l{  }      repeat length times
   «          put the 1st letter at the end
    :         duplicate
      «     put the 1st letter at the end (as the last thing called is duplicate)
       ¹    wrap the stack in an array
        ╬,  quad-palindromize with 1 X and Y overlap

J, 48 bytes

(}:@|.,])(],.}.@|."1)(1+i._26)|."0 1 a.{~97+i.26

Try it online!

Pyth, 21 19 bytes

j+PKm+PJ.<Gd_JS27_K

Try it online!

Explanation:

j+PKm+PJ.<Gd_JS27_K   expects no input

j                     joins on new line
 +   +                joins two strings
  P   P               prints everything but the last element
   K                  initialize K and implicitly print
    m                 for...in loop, uses d as iterator variable
       J              initialize J and implicitly print
        .<            cyclically rotate
          G           initialized to the lowercase alphabet
           d          iterating variables of m
            _    _    reverse
             J        call J
              S27     indexed range of 27
                  K   call K

Jelly, 11 bytes

27RØaṙŒḄŒBY

Explanation:

27R          range of 1...27
   Øa        the alphabet
     ṙ       rotate
      ŒḄŒB   bounce in both dimensions
          Y  join on newline

Racket 293 bytes

(let*((ls list->string)(rr reverse)(sr(λ(s)(ls(rr(string->list s))))))(let p((s(ls(for/list((i(range 97 123)))(integer->char i))))
(n 0)(ol'()))(let*((c(string-ref s 0))(ss(substring s 1 26))(s(string-append ss(string c)(sr ss))))(if(< n 53)(p s(+ 1 n)(cons s ol))
(append(rr ol)(cdr ol))))))

Ungolfed:

(define (f)
  (define (sr s)           ; sub-fn reverse string;
    (list->string
     (reverse
      (string->list s))))
  (let loop ((s
              (list->string
               (for/list
                   ((i
                     (range 97 123)))
                 (integer->char i))))
             (n 0)
             (ol '()))
    (define c (string-ref s 0))
    (define ss (substring s 1 26))
    (set! s (string-append
             ss 
             (string c)
             (sr ss)))
    (if (< n 53)
        (loop s (add1 n) (cons s ol))
        (append (reverse ol) (rest ol)))))

Testing:

(f)

Output:

'("bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb"
  "cdefghijklmnopqrstuvwxyzabazyxwvutsrqponmlkjihgfedc"
  "defghijklmnopqrstuvwxyzabcbazyxwvutsrqponmlkjihgfed"
  "efghijklmnopqrstuvwxyzabcdcbazyxwvutsrqponmlkjihgfe"
  "fghijklmnopqrstuvwxyzabcdedcbazyxwvutsrqponmlkjihgf"
  "ghijklmnopqrstuvwxyzabcdefedcbazyxwvutsrqponmlkjihg"
  "hijklmnopqrstuvwxyzabcdefgfedcbazyxwvutsrqponmlkjih"
  "ijklmnopqrstuvwxyzabcdefghgfedcbazyxwvutsrqponmlkji"
  "jklmnopqrstuvwxyzabcdefghihgfedcbazyxwvutsrqponmlkj"
  "klmnopqrstuvwxyzabcdefghijihgfedcbazyxwvutsrqponmlk"
  "lmnopqrstuvwxyzabcdefghijkjihgfedcbazyxwvutsrqponml"
  "mnopqrstuvwxyzabcdefghijklkjihgfedcbazyxwvutsrqponm"
  "nopqrstuvwxyzabcdefghijklmlkjihgfedcbazyxwvutsrqpon"
  "opqrstuvwxyzabcdefghijklmnmlkjihgfedcbazyxwvutsrqpo"
  "pqrstuvwxyzabcdefghijklmnonmlkjihgfedcbazyxwvutsrqp"
  "qrstuvwxyzabcdefghijklmnoponmlkjihgfedcbazyxwvutsrq"
  "rstuvwxyzabcdefghijklmnopqponmlkjihgfedcbazyxwvutsr"
  "stuvwxyzabcdefghijklmnopqrqponmlkjihgfedcbazyxwvuts"
  "tuvwxyzabcdefghijklmnopqrsrqponmlkjihgfedcbazyxwvut"
  "uvwxyzabcdefghijklmnopqrstsrqponmlkjihgfedcbazyxwvu"
  "vwxyzabcdefghijklmnopqrstutsrqponmlkjihgfedcbazyxwv"
  "wxyzabcdefghijklmnopqrstuvutsrqponmlkjihgfedcbazyxw"
  "xyzabcdefghijklmnopqrstuvwvutsrqponmlkjihgfedcbazyx"
  "yzabcdefghijklmnopqrstuvwxwvutsrqponmlkjihgfedcbazy"
  "zabcdefghijklmnopqrstuvwxyxwvutsrqponmlkjihgfedcbaz"
  "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba"
  "bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb"
  "cdefghijklmnopqrstuvwxyzabazyxwvutsrqponmlkjihgfedc"
  "defghijklmnopqrstuvwxyzabcbazyxwvutsrqponmlkjihgfed"
  "efghijklmnopqrstuvwxyzabcdcbazyxwvutsrqponmlkjihgfe"
  "fghijklmnopqrstuvwxyzabcdedcbazyxwvutsrqponmlkjihgf"
  "ghijklmnopqrstuvwxyzabcdefedcbazyxwvutsrqponmlkjihg"
  "hijklmnopqrstuvwxyzabcdefgfedcbazyxwvutsrqponmlkjih"
  "ijklmnopqrstuvwxyzabcdefghgfedcbazyxwvutsrqponmlkji"
  "jklmnopqrstuvwxyzabcdefghihgfedcbazyxwvutsrqponmlkj"
  "klmnopqrstuvwxyzabcdefghijihgfedcbazyxwvutsrqponmlk"
  "lmnopqrstuvwxyzabcdefghijkjihgfedcbazyxwvutsrqponml"
  "mnopqrstuvwxyzabcdefghijklkjihgfedcbazyxwvutsrqponm"
  "nopqrstuvwxyzabcdefghijklmlkjihgfedcbazyxwvutsrqpon"
  "opqrstuvwxyzabcdefghijklmnmlkjihgfedcbazyxwvutsrqpo"
  "pqrstuvwxyzabcdefghijklmnonmlkjihgfedcbazyxwvutsrqp"
  "qrstuvwxyzabcdefghijklmnoponmlkjihgfedcbazyxwvutsrq"
  "rstuvwxyzabcdefghijklmnopqponmlkjihgfedcbazyxwvutsr"
  "stuvwxyzabcdefghijklmnopqrqponmlkjihgfedcbazyxwvuts"
  "tuvwxyzabcdefghijklmnopqrsrqponmlkjihgfedcbazyxwvut"
  "uvwxyzabcdefghijklmnopqrstsrqponmlkjihgfedcbazyxwvu"
  "vwxyzabcdefghijklmnopqrstutsrqponmlkjihgfedcbazyxwv"
  "wxyzabcdefghijklmnopqrstuvutsrqponmlkjihgfedcbazyxw"
  "xyzabcdefghijklmnopqrstuvwvutsrqponmlkjihgfedcbazyx"
  "yzabcdefghijklmnopqrstuvwxwvutsrqponmlkjihgfedcbazy"
  "zabcdefghijklmnopqrstuvwxyxwvutsrqponmlkjihgfedcbaz"
  "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba"
  "bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb"
  "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba"
  "zabcdefghijklmnopqrstuvwxyxwvutsrqponmlkjihgfedcbaz"
  "yzabcdefghijklmnopqrstuvwxwvutsrqponmlkjihgfedcbazy"
  "xyzabcdefghijklmnopqrstuvwvutsrqponmlkjihgfedcbazyx"
  "wxyzabcdefghijklmnopqrstuvutsrqponmlkjihgfedcbazyxw"
  "vwxyzabcdefghijklmnopqrstutsrqponmlkjihgfedcbazyxwv"
  "uvwxyzabcdefghijklmnopqrstsrqponmlkjihgfedcbazyxwvu"
  "tuvwxyzabcdefghijklmnopqrsrqponmlkjihgfedcbazyxwvut"
  "stuvwxyzabcdefghijklmnopqrqponmlkjihgfedcbazyxwvuts"
  "rstuvwxyzabcdefghijklmnopqponmlkjihgfedcbazyxwvutsr"
  "qrstuvwxyzabcdefghijklmnoponmlkjihgfedcbazyxwvutsrq"
  "pqrstuvwxyzabcdefghijklmnonmlkjihgfedcbazyxwvutsrqp"
  "opqrstuvwxyzabcdefghijklmnmlkjihgfedcbazyxwvutsrqpo"
  "nopqrstuvwxyzabcdefghijklmlkjihgfedcbazyxwvutsrqpon"
  "mnopqrstuvwxyzabcdefghijklkjihgfedcbazyxwvutsrqponm"
  "lmnopqrstuvwxyzabcdefghijkjihgfedcbazyxwvutsrqponml"
  "klmnopqrstuvwxyzabcdefghijihgfedcbazyxwvutsrqponmlk"
  "jklmnopqrstuvwxyzabcdefghihgfedcbazyxwvutsrqponmlkj"
  "ijklmnopqrstuvwxyzabcdefghgfedcbazyxwvutsrqponmlkji"
  "hijklmnopqrstuvwxyzabcdefgfedcbazyxwvutsrqponmlkjih"
  "ghijklmnopqrstuvwxyzabcdefedcbazyxwvutsrqponmlkjihg"
  "fghijklmnopqrstuvwxyzabcdedcbazyxwvutsrqponmlkjihgf"
  "efghijklmnopqrstuvwxyzabcdcbazyxwvutsrqponmlkjihgfe"
  "defghijklmnopqrstuvwxyzabcbazyxwvutsrqponmlkjihgfed"
  "cdefghijklmnopqrstuvwxyzabazyxwvutsrqponmlkjihgfedc"
  "bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb"
  "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba"
  "zabcdefghijklmnopqrstuvwxyxwvutsrqponmlkjihgfedcbaz"
  "yzabcdefghijklmnopqrstuvwxwvutsrqponmlkjihgfedcbazy"
  "xyzabcdefghijklmnopqrstuvwvutsrqponmlkjihgfedcbazyx"
  "wxyzabcdefghijklmnopqrstuvutsrqponmlkjihgfedcbazyxw"
  "vwxyzabcdefghijklmnopqrstutsrqponmlkjihgfedcbazyxwv"
  "uvwxyzabcdefghijklmnopqrstsrqponmlkjihgfedcbazyxwvu"
  "tuvwxyzabcdefghijklmnopqrsrqponmlkjihgfedcbazyxwvut"
  "stuvwxyzabcdefghijklmnopqrqponmlkjihgfedcbazyxwvuts"
  "rstuvwxyzabcdefghijklmnopqponmlkjihgfedcbazyxwvutsr"
  "qrstuvwxyzabcdefghijklmnoponmlkjihgfedcbazyxwvutsrq"
  "pqrstuvwxyzabcdefghijklmnonmlkjihgfedcbazyxwvutsrqp"
  "opqrstuvwxyzabcdefghijklmnmlkjihgfedcbazyxwvutsrqpo"
  "nopqrstuvwxyzabcdefghijklmlkjihgfedcbazyxwvutsrqpon"
  "mnopqrstuvwxyzabcdefghijklkjihgfedcbazyxwvutsrqponm"
  "lmnopqrstuvwxyzabcdefghijkjihgfedcbazyxwvutsrqponml"
  "klmnopqrstuvwxyzabcdefghijihgfedcbazyxwvutsrqponmlk"
  "jklmnopqrstuvwxyzabcdefghihgfedcbazyxwvutsrqponmlkj"
  "ijklmnopqrstuvwxyzabcdefghgfedcbazyxwvutsrqponmlkji"
  "hijklmnopqrstuvwxyzabcdefgfedcbazyxwvutsrqponmlkjih"
  "ghijklmnopqrstuvwxyzabcdefedcbazyxwvutsrqponmlkjihg"
  "fghijklmnopqrstuvwxyzabcdedcbazyxwvutsrqponmlkjihgf"
  "efghijklmnopqrstuvwxyzabcdcbazyxwvutsrqponmlkjihgfe"
  "defghijklmnopqrstuvwxyzabcbazyxwvutsrqponmlkjihgfed"
  "cdefghijklmnopqrstuvwxyzabazyxwvutsrqponmlkjihgfedc"
  "bcdefghijklmnopqrstuvwxyzazyxwvutsrqponmlkjihgfedcb")

Batch, 255 bytes

@echo off
set l=abcdefghijklmnopqrstuvwxyz
set r=yxwvutsrqponmlkjihgfedcba
for /l %%i in (1,1,27)do call:u
for /l %%i in (1,1,25)do call:l
:l
set r=%r:~2%%l:~-1%.
set l=%l:~-2%%l:~0,-2%
:u
set r=%l:~-1%%r:~0,-1%
set l=%l:~1%%l:~0,1%
echo %l%%r%

Explanation: The subroutine u rotates the alphabet outwards by one letter from the centre, which is the pattern used in the top half of the desired output. The subroutine l rotates the alphabet inwards by two letters. It then falls through into the u subroutine, achieving an effective single letter inward rotation. Finally the last line is printed by allowing the code to fall through into the l subroutine.

Groovy - 103 97 bytes

I realise there are cleverer ways of doing this but...

{t=('a'..'z').join();q={it[-2..0]};c=[];27.times{t=t[1..-1]+t[0];c<<t+q(t)};(c+q(c)).join('\n')}

When run the result of the script is the requested answer.

(Thanks to carusocomputing for the tip on saving 7 bytes).

Updated example accordingly on:

See http://ideone.com/MkQeoW

C, 252 bytes

#define j q[y]
#define k(w,v) (v<'z')?(w=v+1):(w=97)
char q[28][52],d;main(){int y,x=1;y=1;j[0]=98;j[50]=98;for(;y<27;y++){for(;x<26;x++){(x<1)?(k(d,q[y-1][50])):(k(d,j[x-1]));j[50-x]=d;j[x]=d;}x=0;j[51]=0;puts(j);}strcpy(j,q[1]);for(;y;y--)puts(j);}

Formatted, macro-expanded version, which is hopefully more intelligible:

#define j q[y]
#define k(w,v) (v<'z')?(w=v+1):(w=97)
char q[28][52],d;
main(){
int y,x=1;
y=1;
q[1][0]=98;q[1][50]=98;
//98 takes one less byte to type than the equivalent 'b'
for(;y<27;y++){
    for(;x<26;x++){
        (x<1)?
            (k(d,q[y-1][50]))
            :(k(d,q[y][x-1]));
        q[y][50-x]=d;
        q[y][x]=d;
        }
    x=0;
    q[y][51]=0;
    puts(q[y]);
    }
strcpy(q[y],q[1]);
for(;y;y--)puts(q[y]);
}

I know this can't win, but I had fun trying. This is my first ever attempt at code golf.

Haskell, 67 66 bytes

unlines[[toEnum$mod(-abs j-abs i)26+97|j<-[-25..25]]|i<-[-26..26]]

Python 3, 119 bytes

I tried to exploit the two symmetry axes of the diamond, but this ended up more verbose than Karl Napf's solution.

A='abcdefghijklmnopqrstuvwxyz'
D=''
for k in range(1,27):
 D+=A[k:]+A[:k]
 D+=D[-2:-27:-1]+'\n'
print(D+D[:51]+D[::-1])

Vim, 62, 60 keystrokes

:se ri|h<_<cr>jjYZZPqqx$pYpq26@qdjyH:g/^/m0<cr>VP:%norm DPA<C-v><C-r>"<C-v><esc>x<cr>

Drawing on inspiration from Lynn's awesome vim answer to take the idea of stealing the alphabet from the help docs.

You can watch it happen in real time as I struggle to remember the right sequence of keystrokes!

Note that this gif is slightly outdated because it produces the wrong output, and I haven't gotten around to re-recording it yet.

JavaScript (ES6), 97 96 bytes

Saved 1 byte thanks to @user81655

R=(n,s=25,c=(n%26+10).toString(36))=>s?c+R(n+1,s-1)+c:c
C=(n=1,r=R(n))=>n<27?r+`
${C(n+1)}
`+r:r

Two recursive functions; C is the one that outputs the correct text. Try it here:

R=(n,s=25,c=(n%26+10).toString(36))=>s?c+R(n+1,s-1)+c:c
C=(n=1,r=R(n))=>n<27?r+`
${C(n+1)}
`+r:r

console.log(C())

PHP, 129 Bytes

for(;++$i<27;)$o.=($s=($f=substr)($r=join(range(a,z)),$i,26-$i)).$t.strrev($s.$t=$f($r,0,$i))."\n";echo$o.$f($o,0,51).strrev($o);

R, 71 bytes

cat(letters[outer(c(1:27,26:1),c(0:25,24:0),"+")%%26+1],sep="",fill=53)

outer creates a matrix with the indices of the letters, letters[...] then creates a vector with the correct letters in. cat(...,sep="",fill=53) then prints it out with the desired formatting.

Perl, 77 bytes

Requires -E at no extra cost.

Pretty standard approach... I don't like the calls to reverse I think there's likely a more maths based approach to this, will see how I get on!

@a=((a..z)x3)[$_..$_+26],$a=pop@a,(say@a,$a,reverse@a)for 1..26,reverse 1..25

Usage

perl -E '@a=((a..z)x3)[$_..$_+26],$a=pop@a,(say@a,$a,reverse@a)for 1..26,reverse 1..25'

05AB1E, 13 12 bytes

A27FÀDûˆ}¯û»

Try it online!

Explanation

A              # push alphabet
 27F           # 27 times do
    À          # rotate alphabet left
     Dû        # create a palendromized copy
       ˆ       # add to global list
        }      # end loop
         ¯     # push global list
          û    # palendromize list
           »   # merge list on newline
               # implicit output

MATL, 14 bytes

2Y226Zv27Zv!+)

Try it online!

2Y2     % Push string 'abc...z'
26Zv    % Push symmetric range [1 2 ... 25 26 25 ... 2 1]
27Zv    % Push symmetric range [1 2 ... 25 26 27 26 25 ... 2 1]
!       % Transpose into a column
+       % Addition with broadcast. Gives a matrix of all pairwise additions:
        % [  2  3 ... 26 27 26 ...  3  2
             3  4 ... 27 28 27 ...  4  3
             ...
            27 28 ... 51 52 51 ... 28 27
            28 29 ... 52 53 52 ... 29 28
            27 28 ... 51 52 51 ... 28 27
             ...
             2  3 ... 26 27 26 ...  3  2 ]
)       % Index modularly into the string. Display implicitly

JavaScript (ES6), 128 115 114 bytes

a='abcdefghijklmnopqrstuvwxyz'
for(o=[i=27];i--;)o[26-i]=o[26+i]=(a=(p=a.slice(1))+a[0])+[...p].reverse().join``
o

Python 2, 96 85 bytes

Printing the uppercase version (saves 1 byte).

R=range
for i in R(53):print''.join(chr(90-(abs(j-25)+abs(i-26)-1)%26)for j in R(51))

previous solution with help from muddyfish

s="abcdefghijklmnopqrstuvwxyz"*3
for i in range(53):j=min(i,52-i);print s[j+1:j+27]+s[j+25:j:-1]

C, 76 bytes

Function, to be called as below. Prints capital letters.

f(i){for(i=2756;--i;)putchar(i%52?90-(abs(i%52-26)+abs(i/52-26)+25)%26:10);}

//call like this
main(){f();}

Simple approach, add the x and y distances from the centre of the square, plus an offset of 25 for the a in the middle, take modulo 26 and subract from 90, the ASCII code for Z. Where i%52==0 a newline ASCII 10 is printed.

Jelly, 13 bytes

Øaṙ1ṭṙJ$ŒBŒḄY

Try it online!

Explanation

Øaṙ1ṭṙJ$ŒBŒḄY  Main link. No arguments
Øa             Get the lowercase alphabet
  ṙ1           Rotate left by 1
    ṭ          Append to
       $       Monadic chain
      J          Indices of the alphabet [1, 2, ..., 26]
     ṙ           Rotate the alphabet by each
        ŒB     Bounce each rotation
          ŒḄ   Bounce the rotations
            Y  Join with newlines and print implicitly

Haskell, 75 bytes

g=(++)<*>reverse.init 
unlines$g$g.take 26.($cycle['a'..'z']).drop<$>[1..27]

How it works:

g=(++)<*>reverse.init    -- helper function that takes a list and appends the
                         -- reverse of the list with the first element dropped, e.g.
                         -- g "abc"  -> "abcba"

             <$>[1..27]  -- map over the list [1..27] the function that
           drop          -- drops that many elements from
    ($cycle['a'..'z'])   -- the infinite cycled alphabet and
   take 26               -- keeps the next 26 chars and
  g                      -- appends the reverse of itself

                         -- now we have the first 27 lines

 g                       -- do another g to append the lower half
unlines                  -- join with newlines

Pyke, 23 22 21 bytes

GV'th+jj_t+j)K]0n'JOX

Try it here!

GV          )         - repeat 26 times, initially push alphabet.
  'th+                -  push tos[1:]+tos[0]
      j               -  j = tos
       j              -  push j
        _t+           -  push pop+reversed(pop)[1:]
           j          -  push j
             K        - pop
              ]0      - list(stack)
                n'JOX - print "\n".join(^),
                      - splat ^[:-1]