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Chess Variant xkcd.com/3139/

You could also have: Wormhole Squares: Pairs of these spaced out over the board - the piece moves out of the Wormhole exactly the same number of squares it needed to move to enter, in the same direction.

Reflective Buffers around the edge - if a piece lands on the square next to one of these, it bounces off the same number of squares in the opposite direction (if a Castle, just goes back, if a Bishop, it bounces at 90°, if a Knight... my brain hurts).

Go To Jail: The piece is taken to the Jail square of the nearest game of Monopoly, where it languishes until a player shows mercy and bails it out.