A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
I believe the answer is No, there is no good classification of these things. It might be helpful for you to know that a "generously transitive permutation group" is the same as a 2-star transitive group. (See for instance the introduction to "The k-star Property for Permutation Groups" by Clough, Praeger, Schneider.) I can email you a copy of this paper if you need it.