Reciprocal, or turn-about, or (other name) chess
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Has anyone seen this before? All captures by Q,R,N,or B of another Q,R,N,or B can be, and can only be done by moving the capturing piece as if it was the same as the piece being captured. For simplicity, captures/checks involving at least one K or P by either side remain the same as standard chess. For example, 1. g3 b6 2. Bg2 Ra8xBg2 (moving like a B).
3. Ng1xRg2 is possible, but instead 3. Nc3 Rg2xNg1 is not allowed because that's a R move, not a N one. In the case of 3. Nf3 Rg1, that's a check on the Ke1 forcing Rh1xRg1, but not Nf3xRg1.