(type A)    (type B)

Looks like a compound of 2 icosidodecahedra (id), and indeed vertices, edges, and either {3} (type A) or {5} (type B) all coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/2x3o3*a (type A)

.   . .    | 60 |  2  2 |  2  1  1
-----------+----+-------+---------
x   . .    |  2 | 60  * |  1  1  0
.   x .    |  2 |  * 60 |  1  0  1
-----------+----+-------+---------
x5/2x .    | 10 |  5  5 | 12  *  *
x   . o3*a |  3 |  3  0 |  * 20  *
.   x3o    |  3 |  0  3 |  *  * 20

x3/2x5o5*a (type B)

.   . .    | 60 |  2  2 |  2  1  1
-----------+----+-------+---------
x   . .    |  2 | 60  * |  1  1  0
.   x .    |  2 |  * 60 |  1  0  1
-----------+----+-------+---------
x3/2x .    |  6 |  3  3 | 20  *  *
x   . o5*a |  5 |  5  0 |  * 12  *
.   x5o    |  5 |  0  5 |  *  * 12

x3/2o3/2x5/2*a (type A)

.   .   .      | 60 |  2  2 |  1  2  1
---------------+----+-------+---------
x   .   .      |  2 | 60  * |  1  1  0
.   .   x      |  2 |  * 60 |  0  1  1
---------------+----+-------+---------
x3/2o   .      |  3 |  3  0 | 20  *  *
x   .   x5/2*a | 10 |  5  5 |  * 12  *
.   o3/2x      |  3 |  0  3 |  *  * 20

x5/4o5/4x3/2*a (type B)

.   .   .      | 60 |  2  2 |  1  2  1
---------------+----+-------+---------
x   .   .      |  2 | 60  * |  1  1  0
.   .   x      |  2 |  * 60 |  0  1  1
---------------+----+-------+---------
x5/4o   .      |  5 |  5  0 | 12  *  *
x   .   x3/2*a |  6 |  3  3 |  * 20  *
.   o5/4x      |  5 |  0  5 |  *  * 12

β3x5o (type B)

both( . . . ) | 60 |  2  2 |  2  1  1
--------------+----+-------+---------
both( . x . ) |  2 | 60  * |  1  1  0
sefa( β3x . ) |  2 |  * 60 |  1  0  1
--------------+----+-------+---------
      β3x .   ♦  6 |  3  3 | 20  *  *
both( . x5o ) |  5 |  5  0 |  * 12  *
sefa( β3x5o ) |  5 |  0  5 |  *  * 12

starting figure: x3x5o

β5x3o (type A)

both( . . . ) | 60 |  2  2 |  2  1  1
--------------+----+-------+---------
both( . x . ) |  2 | 60  * |  1  1  0
sefa( β5x . ) |  2 |  * 60 |  1  0  1
--------------+----+-------+---------
      β5x .   ♦ 10 |  5  5 | 12  *  *
both( . x3o ) |  3 |  3  0 |  * 20  *
sefa( β5x3o ) |  3 |  0  3 |  *  * 20

starting figure: x5x3o