Looks like a compound of 2 great stellated dodecahedra (gissid), and indeed edges and faces both coincide by pairs, but vertices are identified.

Incidence matrix according to Dynkin symbol

o5/3x5/2o3*a

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

o3/2o5/2x5/2*a

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

o3/2o5/3x5/3*a

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