Single Comment

The bent riders and lame leapers (like the Mao) are part of a larger class
that might be called 'multi-movers'.  These are pieces that can make two
(or more) geometrically different moves in the same turn.  The gryphon
moves as a Ferz and then (optionally) as a Rook; the Mao moves as a Wazir
then (mandatorily) as a Ferz. Any such piece will have the asymmetric
retreat property if the order of move types is not reversible. If the
gryphon could move Ferz then Rook or Rook then Ferz it would not have the
asymetric retreat property (and would be immensely powerful).

True leapers such as the Knight in a sense might be said to have the
asymmetric retreat property but it is irrelevant as they can jump over
occupied squares--I prefer to think of a leaper's move as a direct
point-to-point move that does not pass over interventing squares, in which
case the Knights retreat is not asymmetic.

I believe that multi-movers are the only type of pieces which have
symmetric movement patterns but asymmetric retreat. (OK everbody, please
prove me wrong if possible!)