What are examples for *convex* polytope $P\subset \Bbb R^d,d\ge 3$ for which holds


* $P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
* all 2-faces of $P$ are 4-gons (not necessarily squares, or rectangles).


I know the $d$-cubes, rhombic dodecahedron and rhombic triacontahedron. Are there any others?


I expect other such a polytope, if at all, then only in $d\ge 4$. Maybe a slight modification of a *neighborly cubical polytope* as constructed by Ziegler [here](https://arxiv.org/abs/math/9812033), but as they are, they have two 2-face orbits.



Are there any more polytopes whose 2-faces are identical 4-gons?
In fact, there are many to be found on Wikipedia under [isogonal figures](https://en.wikipedia.org/wiki/Isohedral_figure), even in three dimensions.


Examples in dimension *four* are obtained as dual polytope of [runcinated 4-simplex](https://en.wikipedia.org/wiki/Runcinated_5-cell#Runcinated_5-cell) or [runcinated 24-cell](https://en.wikipedia.org/wiki/Runcinated_24-cells). This works, because these both runcinations are


1. edge-transitive, and
2. each edge is contained in exactly four facets.


Since in the dual edges become 2-faces and facets become vertices, the resulting dual polytopes will be 2-face-transitive, and each 2-face will have four vertices, that is, they will be 4-gons.


If I am not mistaken, the cells of these two examples are [trigonal trapezohedron](https://en.wikipedia.org/wiki/Trigonal_trapezohedron) (aka. elongated cubes), and these are 3-dimensional examples also found in the Wikipedia list mentioned above.


