Looks like a compound of 2 truncated great dodecahedra (tigid), and indeed vertices, edges, and {10} all coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/4x5x5*a

.   . .    | 120 |  1  1  1 |  1  1  1
-----------+-----+----------+---------
x   . .    |   2 | 60  *  * |  1  1  0
.   x .    |   2 |  * 60  * |  1  0  1
.   . x    |   2 |  *  * 60 |  0  1  1
-----------+-----+----------+---------
x5/4x .    |  10 |  5  5  * | 12  *  *
x   . x5*a |  10 |  5  0  5 |  * 12  *
.   x5x    |  10 |  0  5  5 |  *  * 12

snubbed forms: s5/4s5s5*a

β5/2x5x

both( .   . . ) | 120 |  1  1  1 |  1  1  1
----------------+-----+----------+---------
both( .   x . ) |   2 | 60  *  * |  1  1  0
both( .   . x ) |   2 |  * 60  * |  1  0  1
sefa( β5/2x . ) |   2 |  *  * 60 |  0  1  1
----------------+-----+----------+---------
both( .   x5x ) |  10 |  5  5  0 | 12  *  *
      β5/2x .   ♦  10 |  5  0  5 |  * 12  *
sefa( β5/2x5x ) |  10 |  0  5  5 |  *  * 12

starting figure: x5/2x5x