Summer of Math Exposition

In chess, neither pawns nor bishops can reach all the squares on the board, and are restricted to a smaller subset of squares - a so-called "colorbinding." Here, we ask: can we classify all the ways that a chess piece (whether that's a piece from ordinary Western chess, a piece from other related chesses like xiangqi and shōgi, or a similar but newly-made-up piece) can be colorbound?