Edge-transitive? See http://en.wikipedia.org/wiki/Edge-transitive_graph
EDIT: Gjergji Zaimi says OP's property implies edge-transitivity, and Bhalchandra D Thatte and Pete L. Clark say that edge-transitivity implies OP's property, so it looks like I made a lucky guess and got the right answer. Doesn't that deserve even one measly upvote?