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]]     

[![37R simple evolution][1]][1]


  [1]: https://i.sstatic.net/1pOPP.png

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.