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This is now ready for editorial review — there are a number of notes/question at the bottom that will need resolving but I'd like the eds' (and any other) input on those.
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I think Angryph, as a contracted version of Anti-gryph, is an excellent name. For the Ski-Angryph I would prefer Hippogryph. Maogryph for the lame version also doesn't sound bad.
As to the mating potential: it cannot have any, as it is a color alternator. This is the reason I chose it as the super-piece in Team-Mate Chess, as no single piece was supposed to have any mating potential there.
Since the Angryph is a very powerful piece, I would be very surprised if it did not have mating potential in combination with any other piece that can access the entire board, no matter how weak. I think it can be proven that a Bishop can force checkmate in combination with any unbound piece, on rectangular boards of any size. For the Angryph, which also covers diagonals, it should be similar. I am not completely sure how this would work out together with a color-bound piece, but on 8x8 an Angryph has no problem checkmating a bare King together with a Ferz (as I just tried out in Fairy-Max).