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Here's an amusing possible solution to the problems with this variant:
combine it with <a href='../other.dir/alice.html'>Alice Chess</a>.
<p>
Here's how it might go. You add a second board, like in Alice Chess,
except the 2nd board has reversed checkering: a1 is white, not black.
When a piece's move would otherwise cause it to move to a square of a
different color, it instead lands on the equivalent square of the
other board. Thus Knights always switch boards when they move, and
Bishops never switch boards.
<p>
There are a number of ways to handle switching boards:
<p>
<ul>
<li>Alice Chess-style. The move on the board on which the piece
starts must be legal as in orthochess, and the square on the other
board must be empty.</li>
<p>
<li>The Plunge. A piece moving to another color may only to move to
a square that is empty on their current board, then they plunge through
the board to the equivalent square on the other board, capturing any
opposing pieces they land on, except for Pawns who may not plunge to
occupied squares.</li>
<p>
<li>The Switch-a-roo. A piece makes a normal orthochess move on the board
on which it starts, and then, if the destination square is of a different
color than the piece's starting square, it moves to an equivalent
position on the other board. If the space on the other board is occupied,
then the piece occupying that space is moved to the space just landed on on
the board that the moving piece started on. This version actually allows
Bishops on the 2nd board.</li>
<p>
<li>The Last Square. The piece's move is as normal, except that if the
piece would land on a color of square different from which it started, the
last square of its move is the equivalent space on the other board, and the
move does not pass through what would be the final square of its move in
orthochess. The last square on the board on which the board-changing piece
moved from may be occupied by a friendly or opposing piece -- it doesn't
matter as the moving piece does not pass through it.
</ul>
<p>
I don't know which would be best.