As abstract polytope cadditradid is automorph, thereby interchanging the roles of pentagons and pentagrams.

Looks like a compound of a ditrigonary dodecadodecahedron (ditdid) plus a rhombihedron (rhom, the compound of 5 cubes), and indeed vertices coincide by three, edges coincide by pairs.

Incidence matrix according to Dynkin symbol

x5/3o5x

.   . . | 60 |  2  2 |  1  2  1
--------+----+-------+---------
x   . . |  2 | 60  * |  1  1  0
.   . x |  2 |  * 60 |  0  1  1
--------+----+-------+---------
x5/3o . |  5 |  5  0 | 12  *  *
x   . x |  4 |  2  2 |  * 30  *
.   o5x |  5 |  0  5 |  *  * 12

x5/4o5/2x

.   .   . | 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   .   x |  4 |  2  2 |  * 30  *
.   o5/2x |  5 |  0  5 |  *  * 12

as uniform compound (type A)

  20 |  6  6 |  3  6  3 || 1 2
-----+-------+----------++----
   2 | 60  * |  1  0  1 || 1 0
   2 |  * 60 |  0  2  0 || 0 1
-----+-------+----------++----
   5 |  5  0 | 12  *  * || 1 0
   4 |  0  4 |  * 30  * || 0 1
   5 |  5  0 |  *  * 12 || 1 0
-----+-------+----------++----
♦ 20 | 60  0 | 20  0 12 || 1 *
♦  8 |  0 12 |  0  6  0 || * 5

as uniform compound (type B)

  20 |  6  6 |  3  6  3 || 1 1
-----+-------+----------++----
   2 | 60  * |  1  0  1 || 1 0
   2 |  * 60 |  0  2  0 || 0 1
-----+-------+----------++----
   5 |  5  0 | 12  *  * || 1 0
   4 |  0  4 |  * 30  * || 0 1
   5 |  5  0 |  *  * 12 || 1 0
-----+-------+----------++----
♦ 20 | 60  0 | 20  0 12 || 1 *
♦ 20 |  0 60 |  0 30  0 || * 1