BQN, 30 28 bytes^SBCS

{∾•Fmt¨×⊸/+˝(⥊𝕩-⌜"aA")=⌜↕26}

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↕26 Range from 0 to 25.
⥊𝕩-⌜"aA" Differences between each of the characters in the input and a or A.
=⌜ Equality table between those two vectors.
+˝ Sum the columns.
×⊸/ Keep the values with sign 1 (or: remove the zeros)
∾•Fmt¨ Convert each value to a string and join.

BQN, 30 28 bytes^SBCS

{∾•Fmt¨×⊸/+˝(⥊𝕩-⌜"aA")=⌜↕26}

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↕26 Range from 0 to 25.
⥊𝕩-⌜"aA" Differences between each of the characters in the input and a or A.
=⌜ Equality table between those two vectors.
+˝ Sum the columns.
×⊸/ Keep the values with sign 1 (or: remove the zeros)
∾•Fmt¨ Convert each value to a string and join.

A slightly different approach using Bins Down at 29 bytes:

{∾•Fmt¨×⊸/»1↓/⁼27↕⊸⍋⥊𝕩-⌜"aA"}

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BQN, 30 bytes^SBCS

{∾•Fmt¨×⊸/+˝(𝕩∾32+𝕩)=⌜'a'+↕26}

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'a'+↕26 The letters from a to z.
𝕩∾32+𝕩 The input string joined with the input string where each codepoint is increased by 32.
=⌜ Equality table between those two strings.
+˝ Sum the columns.
×⊸/ Keep the values with sign 1 (or: remove the zeros)
∾•Fmt¨ Convert each value to a string and join.

BQN, 30 28 bytes^SBCS

{∾•Fmt¨×⊸/+˝(⥊𝕩-⌜"aA")=⌜↕26}

Run online!

↕26 Range from 0 to 25.
⥊𝕩-⌜"aA" Differences between each of the characters in the input and a or A.
=⌜ Equality table between those two vectors.
+˝ Sum the columns.
×⊸/ Keep the values with sign 1 (or: remove the zeros)
∾•Fmt¨ Convert each value to a string and join.