A polyhedron is quasi-quasi-regular if it is vertex-transitive with
trapezoidal vertex figures. Vertex transitivity means that for any two
vertices of the polyhedron, there exists a translation, rotation, and/or
reflection that leaves the outward appearance of the polyhedron unchanged
yet moves one vertex to the other. A vertex figure is the polygon produced
by connecting the midpoints of the edges meeting at the vertex in the same
order that the edges appear around the vertex. There are only two
quasi-quasi-regular polyhedra that are not self-intersecting, namely the
Rhombicuboctahedron
and the Rhombicosidodecahedron.
When self-intersection is allowed, there are 12 other quasi-quasi-regular
polyhedra.

Albert Badoureau described 11 of these polyhedra (all except the Small
Ditrigonal Dodecicosidodecahedron) in 1881
[1].
Johann Pitsch described five (the Great Cubicuboctahedron, the
Great Dodecicosidodecahedron, the Great Ditrigonal Dodecicosidodecahedron,
the Small Icosicosidodecahedron, and the Rhombidodecadodecahedron)
in 1881 [2].
H. S. M. Coxeter and J. C. P. Miller discovered the Small Ditrigonal
Dodecicosidodecahedron between 1930 and 1932
[3].

Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique49 (1881), 47-172.

[2]

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

[3]

H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences246 (1954), 401-450.