A polyhedron is versi-quasi-regular if it is vertex-transitive with crossed
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 seven
versi-quasi-regular polyhedra, all of which are self-intersecting.
All seven have non-orientable surfaces (like that of a
Klein Bottle
or the
Real Projective
Plane).

Albert Badoureau described six of these polyhedra (all except the Small
Dodecicosahedron) in 1881 [1].
H. S. M. Coxeter and J. C. P. Miller discovered the Small Dodecicosahedron
between 1930 and 1932 [2].

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

[2]

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.