Returning to the Octagonal analogue to the Mao, it occurs to me that there are so many possibilities for two successive "outward" moves that there should be a general term. With Gryphon being the oldest example of a two-stage piece and my extrapolations including the Fury, Gorgon, Harpy, Hydra, Lamia, Manticore, Simurgh, and Sphinx I wondered about the first and second stage hyphenated (as with hunter and sniper) followed by monster. Thus the Gryphon would be a Ferz-Rook monster, the Simurgh a Viceroy-Rook monster, and the Contragryphon a Rook-Ferz monster. To give examples with a non-monster name the Anchorite would be a Wazir-Bishop monster and the Farrier a Wazir-Unicorn monster. The compound of two monsters with the same first stage would be thar stage's monster with the compound of the second stages and vice versa, making the Ostler a Wazir-Governor monster and the Rooksheath a Baron-Rook monster. Daniil Frolov's piece is the compound of a Wazir-Knight monster and Ferz-Knight monster, and so a Prince-Knight monster. What would however need a formal definition is exactly what "outward" means.
The best that I can devise is that at the turning point it might go through a radial but not bounce off one. Here are some examples on the FIDE board: Valid Prince-Knight Monster moves include a1-a2-b4, going through the a2-b1 and a2-g8 diagonals; a1-b1-d2, going through the a2-b1 and b1-h7 diagonals; and a1-b2-c4 and a1-b2-d3, both going through rank 2 and file b. They do not include a1-a2-c3, as it bounces off the a2-g8 diagonal; a1-b1-c3, as it bounces off the b1-h7 diagonal; a1-b2-a4, as it bounces off file b; or a1-b2-d1, as it bounces off rank 2. Does this seem a satisfactory and sufficiently rigorous definition?
The best that I can devise is that at the turning point it might go through a radial but not bounce off one. Here are some examples on the FIDE board: Valid Prince-Knight Monster moves include a1-a2-b4, going through the a2-b1 and a2-g8 diagonals; a1-b1-d2, going through the a2-b1 and b1-h7 diagonals; and a1-b2-c4 and a1-b2-d3, both going through rank 2 and file b. They do not include a1-a2-c3, as it bounces off the a2-g8 diagonal; a1-b1-c3, as it bounces off the b1-h7 diagonal; a1-b2-a4, as it bounces off file b; or a1-b2-d1, as it bounces off rank 2. Does this seem a satisfactory and sufficiently rigorous definition?