The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) involutions, or (B) they are each other's inverses and of order >2. Note that

- non-isomorphic Cayley graphs can be isomorphic as graphs, so that the same cubic graph may arise as a Cayley graph of both types A and B;
- a single group can have non-isomorphic cubic Cayley graphs, so that the same group may have Cayley graphs of both types A and B;
- in particular, it might even be possible for a single group to give rise to non-isomorphic cubic Cayley graphs which are isomorphic as graphs.

What is known about the characterizations of, or relationship between, the two types A and B of cubic Cayley graphs, and in particular the situations in which they can coincide - either in the sense of being isomorphic as graphs, or of coming from the same group (or possibly both)?

I hope this question is not too vague.