Looks like a compound of 2 tetrahemihexahedra (thah), and indeed vertices, edges, and faces all coincide by pairs. Each individual vertex figure is the same as the vertex figure of thah, a crossed square.

Incidence matrix according to Dynkin symbol

x3/2o3x

.   . . | 12 |  2  2 | 1 2 1
--------+----+-------+------
x   . . |  2 | 12  * | 1 1 0
.   . x |  2 |  * 12 | 0 1 1
--------+----+-------+------
x3/2o . |  3 |  3  0 | 4 * *
x   . x |  4 |  2  2 | * 6 *
    o3x |  3 |  0  3 | * * 4

snubbed forms: β3/2o3x, x3/2o3β, β3/2o3β