TITLE: “Totally transitive” polytopes which are not regular
QUESTION [8 upvotes]: Is it possible to have an abstract polytope which is vertex-transitive, edge-transitive, face-transitive, etc. (individually transitive on faces of each particular dimension) and yet not flag-transitive?
[I had thought this might be an easy curiosity for Math StackExchange to readily answer for me, but I had no engagement there, and thus I now cross-post it here…]

REPLY [3 votes]: An illustration for the answer by John Machacek:

Identifying opposite sides of the picture, one gets a 3-polytope with five vertices, ten edges and five square faces. (This is the well known torus map $\{4,4\}_{(1,2)}$.)