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This is in principle not different from any other variant where there are pieces that break left-right symmetry: You can give the opponents the same or the reflected pieces. E.g. on a square board a right-handed 'Skewed Rook' which moves N-S like a Rook, and NE-SW like a Bishop. You can then make the corresponding piece of the opponent move NE-SW as well, or NW-SE (besides N-S).
The difference from a normal FIDE-pieces-versus-non-FIDE-pieces variant is that both players can visualise their own pieces as FIDE ones - and, by looking at the board in a slightly different way, their opponents' ones as well. They don't have to think of one side's 'Rooks' as compounds of a Rookfiler and half a Bishop, and 'Bishops' as filebound compounds of a Rookranker and a quarter of a Nightrider, and 'Knights' as fileswitching compounds of half a Ferz and a quarter of a Buffalo! They don't even have to notice - as I did only while plotting out these moves - that one player's division into odd and even ranks is the other player's Bishop colouring.
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but what about one that White sees as:
and Black as
where upright pieces represent standard FIDE ones and sideways ones something more exotic? Do you understand the idea?