Like the Archbishop, which it resembles, the exact origins of this piece are unknown (if you have more information, please let us know). The earliest reference I have to it is Billiards Chess, which was invented by M. Jacques Berthoumeau in the 1950's, but the piece was most likely not new then.
The Reflecting Bishop is a fairly popular piece, and is used in a number of games, including: Billards Chess, Billards Progressive Chess, Edgehog Chess, Knighrider Bouncy Chess, Lambeth Conference Chess and Squarcle Chess.
The Reflecting Bishop moves like a normal chess Bishop, but it has the additional power of reflecting off the side of the board at a 90 degree angle. It may not move to the square on which it starts its move.
In the diagram below, the Reflecting Bishop, which is traditionally symbolized by an upside-down bishop, can move to all the squares marked with a black circle.