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].