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

Incidence matrix according to Dynkin symbol

x5/4o5o5*a

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
x   . .    |  2 | 60 |  1  1
-----------+----+----+------
x5/4o .    |  5 |  5 | 12  *
x   . o5*a |  5 |  5 |  * 12

o5/4o5x5*a

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

x5/4o5/4o5/4*a

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
x   .   .      |  2 | 60 |  1  1
---------------+----+----+------
x5/4o   .      |  5 |  5 | 12  *
x   .   o5/4*a |  5 |  5 |  * 12

β5/2o5o

both( .   . . ) | 12 | 10 |  5  5
----------------+----+----+------
sefa( β5/2o . ) |  2 | 60 |  1  1
----------------+----+----+------
      β5/2o .   ♦  5 |  5 | 12  *
sefa( β5/2o5o ) |  5 |  5 |  * 12

starting figure: x5/2o5o