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My s[w]eeping switchers army for chess with different
armies features three pieces with assymetric retreat.
All of them are bent riders: The panda (aka slip rook),
the erl queen (aka slip queen) and the unicorn.

Another one is the mao (xiangi horse) which is not a bent
rider.

--Jörg Knappen

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!)

well, besides multi-movers, leaping pieces such as grasshoppers have
symmetric move patterns but assymetric retreat.  and xiangqi cannon has
assymetric retreat when capturing but not when moving, which is one of the
things that makes it such a neat piece (and difficult to get used to).<P>

and incidentally any piece that move differently forwards than backwards
(these pieces don't have symmetric move patterns, at least not about the
x-axis) is going to have assymetric retreat.  this includes lots of betza
pieces such as fBbR, fRbB, etc etc (i could go on and on) and shogi pieces
(which can of course be easily described in betza notation) such as the
gold, silver, copper generals, the white horse and the whale, etc etc.

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.

Thank you all for your replies.

Might be interesting to have a large variant built on the general theme of
asymmetry: some pieces would have the asymmetric retreat property but have
symetric move patterns, some would also have asymmetric forward and
backward moves, some with asymmetric left and right moves, some with
divergent captures, etc. Perhaps a 11 by 11 game with strong pieces and a
strong, asymmetric King.

11 by 11?  Shouldn't a game with an asymmetry theme be played on an
asymmetrical board? :-)

Perhaps the 43-square contest would be perfect for an Asymmetry Chess...

I like the idea of the large board.

In addition to the general game with the ability to select armies, I feel
that it might be nice to define a game with preset armies and a
distinctive name as a particular case of the general format so that a
collection of games could hopefully be produced by various players.