Alphabet Diamond

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Note that this gif is slightly outdated because it produces the wrong output, and I haven't gotten around to re-recording it yet.

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