The dual of a quasi-regular polyhedron is face-transitive and edge-transitive
but not vertex-transitive. In this context, transitivity means that for any
two faces (edges, vertices) of the polyhedron, there exists a translation,
rotation, and/or reflection that leaves the outward appearance of the
polyhedron unchanged yet moves one face (edge, vertex) to the other.
This definition implies that a quasi-regular dual must have two kinds of
regular vertices, where each vertex of one type is connected along all of its
edges to vertices of the other type. There are only two quasi-regular duals
that are not self-intersecting, namely the
Rhombic Dodecahedron and the
Rhombic Triacontahedron.
When self-intersection is allowed, there are 14 other quasi-regular duals,
but only five of these are finite.
The nine versi-regular polyhedra, due to their
faces that pass through their centers, produce duals with vertices at infinity.

Edmund Hess, Über vier Archimedeische Polyeder höherer Art, Schriften der Gesellschaft zur Beförderung der gesammten Naturwissenschaften zu Marburg11(4) (1878).

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Johann Pitsch, Über Halbreguläre Sternpolyeder, Zeitschrift für das Realschulwesen6 (1881), 9-24, 64-65, 72-89, 216.