If I understand what a graph automorphism is, what I want is a name for the graphs that satisfy:
For each vertex pair (u, v) there is an automorphism f such that f(u) = v.
A symmetric graph is not exactly that, and I'm a bit confused. https://en.wikipedia.org/wiki/Symmetric_graph
"vertex-transitive" is what I was looking for. Woohoo. https://en.wikipedia.org/wiki/Vertex-transitive_graph