Looks like a compound of 2 icosahedra (ike), and indeed edges and faces both coincide by pairs, but vertices are identified.

As abstract polytope 2ike is isomorphic to 2gike.

Incidence matrix according to Dynkin symbol

o5/2o3x3*a

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x3*a |  3 |  3 |  * 20

x3/2o5/3o3*a

.   .   .    | 12 | 10 |  5  5
-------------+----+----+------
x   .   .    |  2 | 60 |  1  1
-------------+----+----+------
x3/2o   .    |  3 |  3 | 20  *
x   .   o3*a |  3 |  3 |  * 20

o3/2x3/2o5/2*a

.   .   . | 12 | 10 |  5  5
----------+----+----+------
.   x   . |  2 | 60 |  1  1
----------+----+----+------
o3/2x   . |  3 |  3 | 20  *
.   x3/2o |  3 |  3 |  * 20