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

Incidence matrix according to Dynkin symbol

o3/2o5x5*a

.   . .    | 20 |  6 |  3  3
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o5x    |  5 |  5 | 12  *
o   . x5*a |  5 |  5 |  * 12

x5/4o3o5*a

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

o5/4x5/4o3/2*a

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