Looks like a compound of 2 small rhombicuboctahedra (sirco), and indeed vertices, edges, and {4} coincide by pairs.

Incidence matrix according to Dynkin symbol

β4β3x

demi( . . . (a)) | 24  * |  1  2  0  1 | 1 0 1  2
demi( . . . (b)) |  * 24 |  1  0  2  1 | 0 1 1  2
-----------------+-------+-------------+---------
both( . . x    ) |  1  1 | 24  *  *  * | 0 0 1  1
sefa( s4s . (a)) |  2  0 |  * 24  *  * | 1 0 0  1
sefa( s4s . (b)) |  0  2 |  *  * 24  * | 0 1 0  1
sefa( . β3x    ) |  1  1 |  *  *  * 24 | 0 0 1  1
-----------------+-------+-------------+---------
      s4s . (a)  ♦  4  0 |  0  4  0  0 | 6 * *  *
      s4s . (b)  ♦  0  4 |  0  0  4  0 | * 6 *  *
      . β3x      ♦  3  3 |  3  0  0  3 | * * 8  *
sefa( β4β3x    ) |  2  2 |  1  1  1  1 | * * * 24

or
both( . . . )    | 48 |  1  2  1 |  1 1  2
-----------------+----+----------+--------
both( . . x )    |  2 | 24  *  * |  0 1  1
sefa( s4s . )  & |  2 |  * 48  * |  1 0  1
sefa( . β3x )    |  2 |  *  * 24 |  0 1  1
-----------------+----+----------+--------
      s4s .    & ♦  4 |  0  4  0 | 12 *  *
      . β3x      ♦  6 |  3  0  3 |  * 8  *
sefa( β4β3x )    |  4 |  1  2  1 |  * * 24

starting figure: x4x3x

x3β4x

demi( . . . (a)) | 24  * |  1  1  0  1  1  0 | 1 1 0  1  1
demi( . . . (b)) |  * 24 |  1  0  1  1  0  1 | 1 0 1  1  1
-----------------+-------+-------------------+------------
both( x . .    ) |  1  1 | 24  *  *  *  *  * | 1 0 0  1  0
demi( . . x (a)) |  2  0 |  * 12  *  *  *  * | 0 1 0  1  0
demi( . . x (b)) |  0  2 |  *  * 12  *  *  * | 0 0 1  1  0
sefa( x3β .    ) |  1  1 |  *  *  * 24  *  * | 1 0 0  0  1
sefa( . s4x (a)) |  2  0 |  *  *  *  * 12  * | 0 1 0  0  1
sefa( . s4x (b)) |  0  2 |  *  *  *  *  * 12 | 0 0 1  0  1
-----------------+-------+-------------------+------------
      x3β .      ♦  3  3 |  3  0  0  3  0  0 | 8 * *  *  *
      . s4x (a)  ♦  4  0 |  0  2  0  0  2  0 | * 6 *  *  *
      . s4x (b)  ♦  0  4 |  0  0  2  0  0  2 | * * 6  *  *
both( x . x    ) |  2  2 |  2  1  1  0  0  0 | * * * 12  *
sefa( x3β4x    ) |  2  2 |  0  0  0  2  1  1 | * * *  * 12

or
both( . . . )    | 48 |  1  1  1  1 | 1  1  1  1
-----------------+----+-------------+-----------
both( x . . )    |  2 | 24  *  *  * | 1  0  1  0
both( . . x )    |  2 |  * 24  *  * | 0  1  1  0
sefa( x3β . )    |  2 |  *  * 24  * | 1  0  0  1
sefa( . s4x )  & |  2 |  *  *  * 24 | 0  1  0  1
-----------------+----+-------------+-----------
      x3β .      ♦  6 |  3  0  3  0 | 8  *  *  *
      . s4x    & ♦  4 |  0  2  0  2 | * 12  *  *
both( x . x )    |  4 |  2  2  0  0 | *  * 12  *
sefa( x3β4x )    |  4 |  0  0  2  2 | *  *  * 12

starting figure: x3x4x