Looks like a compound of 2 small dodecahemidodecahedron (sidhid), and indeed vertices, edges, and faces all coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/4o5x5*a

.   . .    | 60 |  2  2 |  1  2  1
-----------+----+-------+---------
x   . .    |  2 | 60  * |  1  1  0
.   . x    |  2 |  * 60 |  0  1  1
-----------+----+-------+---------
x5/4o .    |  5 |  5  0 | 12  *  *
x   . x5*a | 10 |  5  5 |  * 12  *
.   o5x    |  5 |  0  5 |  *  * 12

β5/2o5x

both( .   . . ) | 60 |  2  2 |  1  1  2
----------------+----+-------+---------
both( .   . x ) |  2 | 60  * |  1  0  1
sefa( β5/2o . ) |  2 |  * 60 |  0  1  1
----------------+----+-------+---------
both( .   o5x ) |  5 |  5  0 | 12  *  *
      β5/2o .   ♦  5 |  0  5 |  * 12  *
sefa( β5/2o5x ) | 10 |  5  5 |  *  * 12

starting figure: x5/2o5x