Numbers of reversible [cellular automaton] rules

For k = 2, r = 1, there are 6 reversible rules, as shown on page 436. For k = 2, r = 2 there are 62 reversible rules, in 20 families inequivalent under symmetries, out of a total of 2^{32} or about 4 billion possible rules. For k = 3, r = 1 there are 1800 reversible rules, in 172 families. For k = 4, r = 1, some of the reversible rules can be constructed from the second-order cellular automata below. Note that for any k and r, no non-trivial totalistic rule can ever be reversible.