A rule R is reversible if there exists a rule R^-1 such that for all a, b, c, d, e, f, g we see the following evolution (where . stands for "don't care"):

A rule R is reversible if there exists a rule R^-1 such that for all a, b, c, d, e, f, g we see the following evolution (where . stands for "don't care"): a b c d e f g Apply rule R . b' c' d' e' f' . Apply rule R^-1 . . c d e . . In equations, it means there must exist some R^-1 such that for all a, b, c, d, e, f, g the following holds: c = R^-1(R(a, b, c), R(b, c, d), R(c, d, e)) d = R^-1(R(b, c, d), R(c, d, e), R(d, e, f)) e = R^-1(R(c, d, e), R(d, e, f), R(e, f, g)) Writing a quick program that checks all possible combinations of R and R^-1 (https://play.rust-lang.org/?version=stable&mode=debug&edition=2021&gist=e677edf47d45067cfa4988b8680c5a79 https://play.rust-lang.org/?version=stable&mode=debug&editio...) we find the following inverse pairs: 0x33 <-> 0x33 0x55 <-> 0x0f 0xcc <-> 0xcc 0xf0 <-> 0xaa That is, the following two rules are self-inverses (they're the NOT and IDENTITY gates on the center cell respectively): 111 110 101 100 011 010 001 000 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 We have the left <-> right moving pair you identified: 111 110 101 100 011 010 001 000 1 0 1 0 1 0 1 0 111 110 101 100 011 010 001 000 1 1 1 1 0 0 0 0 And there's just one more, which is the same as the above but it also inverts the output (move left and invert is the inverse of move right and invert): 111 110 101 100 011 010 001 000 0 1 0 1 0 1 0 1 111 110 101 100 011 010 001 000 0 0 0 0 1 1 1 1