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I always felt the simplest chess variant, which basically doesn't affect the game except for throwing out the opening book, is one where we swap white's queen and king (but don't move black's queen and king) and make the corresponding change in castling rules.
This one one of the possible Displacement chess opening setups.
Thanks for the info on Mir Chess 32 and Midgard Chess. I'll add them to the list of games with reverse symmetry.
Another notable variant with reverse symmetry is Wildebeest Chess. It is interesting that, relative to Chaturanga, FIDE Chess removed reverse symmetry while also adding colourbound and compound analogues to the Rook, and Wildebeest Chess restored it while also adding corresponding analogues to the Knight.
Reverse symmetry gives less of an advantage to the first or second player, while making the game more aggressive with Castling. It's also easier to remember openings when you switch sides.
Modern Ministers Courier Chess is the most recent variant to join the Reverse Symmetric group.
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Rotational Symmetry is another name to describe these setups, where a piece at White's right hand will be matched by a corresponding piece at Black's right hand. The Burmese game of Sittuyin is the oldest example I know where the initial Pawn formations are staggered. I have used this idea in my 8x8 variants Mir Chess 32 and Midgard Chess.