> My question is how can we test which of the 256 rules are reversible and how do they pair up?

I wondered this too. It feel like a generalization of the well-studied deconvolution problem, but thankfully without noise.

I haven't had my coffee yet, but some quick googling any thinking didn't deliver an answer, so I'd just try enumerating all the rules and seeing which ones invert each other.

Or, take all permutations of five bits, and find the central three bits in the next generation. If-and-only-if all patterns resulting in the three bits have the same central bit in the original pattern, then the rule is invertible.