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Your interactive diagram has given the pieces too much power: a piece never changes both its board and its relative position. (I've described this before as "2+2 dimensions" as opposed to a fully symmetric 4 dimensions.)
Yours still seems to overprescribe movements (but this has been wonderful as a start to more-symmetric 4d games like my own TessChess, for which I'm dabbling with expanding your diagram here). Here's my attempt. Let me know if you have questions about the rules, or if I seem to have Betza'ed incorrectly (likely).
Oh, I see.
I thought there would be no AYY because I thought "they change either the letter or number (but not both)" meant no moving diagonally between the boards & somehow I missed that moving diagonally doesn't change the number.
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Not Sphinx Chess 960
Edit: What should I call this chess variant I invented because I misinterpreted the rules?