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Thanks, Ben.  Cannon type pieces of course have asymmetic retreat (though
these could be arguably defined as a subtype of multi-movers). Indeed the
Grasshopper and the Cannon when capturing have a stronger form of
asymettric retreat.  Some definitions:

1. High-power Symmetric Retreat--the piece can alway return to its
starting square on the next move by reversing its path (unless prohibited
by the need to meet check, etc.) Example:  Knight. Nb1-c3 can always be
followed by Nc3-b1.

2. Low-power Symmetric Retreat--the piece can return to its starting
square on the next move by reversing its path unless the opponent has used
his turn to block it.
Example: Rook.  Rc3-h3 can be follewd by Rh3-c3 if opponent has not moved
a piece to d3,e3,f3 or g3.

3. Low-power Asymmetric Retreat--the piece cannot reverse its path but may
be able to return to its starting square on the next move if the alternate
retreat path is not blocked. Example: Gryphon.

4. High-power Asymmetric Retreat--the piece cannot return to its starting
square on the next move unless the opponent moves to facilitate it.
Example: Grasshopper.  Gc3-c7 cannot be followed by Gc7-c3 unless the
opponent moves a piece to c4.