The Rook moves as it does in other spherical variants, such as Miller's, Nadvorney's, and Chou's Chess on the Dot.
The Knight moves as it does in Nadvorney's Spherical Chess, which is the same as it moves in Chess on the Dot.
The Bishop moves as it does in Chao's Chess on the Dot, though there are some details I'm not sure about yet.
The Queen moves as a Rook or a Bishop.
In addition to moving one space as a Queen, a King near a pole can make two moves that are unavailable to Rooks, Bishops, and Queens. It can move to the space on either side of the one directly opposite it across the pole.
I have been testing the ZRF, which I got on the Zillions-of-Games website here:
https://zillions-of-games.com/cgi-bin/zilligames/submissions.cgi?do=show;id=450
It looks like the following may be true: