The root 'schizo' means 'split', and the connotation of schizophrenia is appropriate, as you end up having to play two games simultaneously in your head. To play Schizochess you will need a chessboard that has been split lengthwise down the middle. A cheap one made of cardboard is easy to split horizontally at the fold. I recommend just rotating it 90 degrees and ignoring the fact that the colors are now incorrect.
Rules Begin play as normal.
After every fifth move, swap the two halves of the board, so that the left edge of the board becomes the right edge of the centerline, and vice-versa. The board before and after a swap of the halves. Note that the swap happens after five moves, not five pairs of moves. A player who is in check and will be taken out of check by a swap after his or her move is not required to do anything else to defend against check. The en passant rule is probably also unaffected, but we never remembered to play with it anyway, so beware.