Is the number of inequivalent elementary cellular automata rules really 88?

Question

Everywhere from Wolfram's "New Kind of Science" (p. 57) to Wikipedia they say that, out of all possible 256 (=2^8) elementary cellular automata rules, 88 are inequivalent (as defined in the Wikipedia article).

Now, the problem is that I am failing to reproduce the 88 number. No matter how I try - I always get 72.

And it gets worse. Sequence A005418 seems to be defined exactly as Wolfram's inequivalence (see the comment: "Number of bit strings of length (n-1), not counting strings which are the end-for-end reversal or the 0-for-1 reversal of each other as different."). And in that sequence at position 8 they also have 72, not 88.

Off course, there is a chance that my program is wrong and/or I do not fully understand the definition of rule equivalence. Therefore I would like to ask someone to reproduce the computations, and either confirm Wolfram's original number 88, or give me the correct number of inequivalent rules (be it 72, or something else).

This problem is important for a research paper I am working on. And it's not very difficult to implement.