Actually I can see more of it now that you've thought of collapsing the
board along a Unicorn line. It is not hard to see that, given that 1:1:0
and 1:-1:0 are at right angles, so are 1:1:1 and 1:-1:0 - the principle on
which Man and Beast 09's Hyperrhino alternator is built. However, it is
equally clearly at right angles to 1:0:-1 and 0:1:-1 - so the three axes of
the Bishop at right angle to any axis of the Unicorn correspond to the
three axes of the Rook on the hex board.