Looks like a compound of 2 great dodecahemicosahedra (gidhei), and indeed vertices, edges, and faces both coincide by pairs.

Incidence matrix according to Dynkin symbol

o5/4x3x5*a

.   . .    | 60 |  2  2 |  1  2  1
-----------+----+-------+---------
.   x .    |  2 | 60  * |  1  1  0
.   . x    |  2 |  * 60 |  0  1  1
-----------+----+-------+---------
o5/4x .    |  5 |  5  0 | 12  *  *
.   x3x    |  6 |  3  3 |  * 20  *
o   . x5*a |  5 |  0  5 |  *  * 12