Imposing reciprocity in the definition of vertex-transitivity

Question

A simple, undirected graph is vertex-transitive if for any pair of vertices $x,y$, there exists an automorphism (adjacency-preserving self-bijection) $\phi$ such that $\phi(x)=y$.

What if, instead of taking $x$ to $y$ as above, we require the automorphism $\phi$ to exchange $x$ and $y$, i.e. $\phi(x)=y$ and $\phi(y)=x$?

1. Is there a name for this natural refinement of the notion of vertex-transitivity?
2. What is a simple example of a vertex-transitive graph which does not satisfy this?

Note that any Cayley graph whose generating set is conjugacy-invariant does satisfy this exchange property (take $\phi(u)=xu^{-1}y$).

Answer 1

A permutation group $G$ acting on a set $X$ is called generously transitive if for any two elements $x$, $y \in X$ there is a permutation $g \in G$ such that $x^g = y$ and $y^g = x$.

It is fairly easy to find examples of vertex-transitive graphs whose automorphism group is not generously transitive, by searching MathOverflow or math.stackexchange.

For example: https://math.stackexchange.com/questions/953856/automorphism-groups-of-vertex-transitive-graphs