Computing Reversible CA with CellularAutomaton

Question

This computes and plots 127 steps of the evolution of 150R (the reversible version of rule 150).

ArrayPlot[
 CellularAutomaton[{Mod[Total[Flatten[#]], 2] &, {}, {{-1, 
     0}, {0, -1}, {0, 0}, {0, 1}}, 2}, {{{1}, {1}}, 0}, 127]]

This example appears as part of the documentation of CellularAutomaton. How can I compute the evolution of other reversible rules, like 37R, 90R, etc.?

Edit: Below is an image showing the transition functions of both rule 150 and 150R.

rule 150

rule 150R

Rule 150R does a XOR between the regular output of rule 150 for $t+1$ and the state of the cell at $t-1$. Other reversible CA have the same bit-flip behavior when $t-1$ is 1.

Answer 1

The higher order CellularAutomaton is not well documented. Anyway, this is my attempt to define 37R automaton. I am not completely sure if it works correctly, but for me, for a while, it works OK.

First I define the function that gives the new cell depending on two 3-cell neighborhoods. The 37R rule function is defined by

    d37 = IntegerDigits[37, 2, 8]
    Clear[f]
    f[1, 1, 1] = d37[[1]];
    f[1, 1, 0] = d37[[2]];
    f[1, 0, 1] = d37[[3]];
    f[1, 0, 0] = d37[[4]];
    f[0, 1, 1] = d37[[5]];
    f[0, 1, 0] = d37[[6]];
    f[0, 0, 1] = d37[[7]];
    f[0, 0, 0] = d37[[8]];
    g[x_] := Mod[f[x[[2]][[1]], x[[2]][[2]], x[[2]][[3]]] + x[[1]][[2]],2]

Here x is a list of the current and of the previous step 3-neighborhoods.Then I can plot the time evolution starting, for instance, from {{0,1,0},{0,1,0}} surrounded by zeros

    ArrayPlot[CellularAutomaton[{g[#] &, {}, 1, 2}, {{{1}, {1}}, 0}, 200]]     

Probably it can be simplified and adapted for other ECA. The rule 150 (with Total) from the documentation is not a good example for a general reversible automaton function.