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Jelly, 4 bytes

œ?ØẠ

A monadic Link accepting an integer which yields a list of 52 characters.

The input range may be up to somewhat more than \$-2^{223} \leq n \lt 2^{223}\$ since \52ドル!>2^{224}\$.

Try it online!

How?

œ?ØẠ - Link: integer, n
 ØẠ - alphabet (Jelly's longest built-in character list containing no digits
 - or hyphen) = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
œ? - permutation at index (n) if all 52! permutations were written out
 - in lexicographical order.
 - This indexing is modular so when n = -x we fetch the (52!-x)th entry.

So...

-2147483648 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONIGCDAMLJKFEHB
-2147483647 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONIGCDAMLJKFHBE
-2147483646 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONIGCDAMLJKFHEB
-2147483645 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONIGCDAMLJKHBEF
 ...
 -4 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDACB
 -3 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDBAC
 -2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDBCA
 -1 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCAB
 0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
 1 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz
 2 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxzy
 3 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwyxz
 4 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwyzx
 ...
 2147483644 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmrtxwznoqpsvyu
 2147483645 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmrtxwznoqpsyuv
 2147483646 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmrtxwznoqpsyvu
 2147483647 ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmrtxwznoqpusvy