Looks like a compound of 2 coincident small rhombated tesseract (srit). And indeed all but the Grünbaumian elements coincide by pairs. It occurs in different types. Type A uses pairs of coincident small rhombicuboctahedra (sirco), while all other cells are doubly covered. Type B uses pairs of coincident triangular prisms (trip), while all other cells are doubly covered.

Incidence matrix according to Dynkin symbol

β3x3o4x   (type A)

both( . . . . ) | 192 |   2   2   2 |  1  2  1  2  1  2 | 1  1  2 1
----------------+-----+-------------+-------------------+----------
both( . x . . ) |   2 | 192   *   * |  1  1  0  1  0  0 | 1  1  1 0
both( . . . x ) |   2 |   * 192   * |  0  1  1  0  0  1 | 1  0  1 1
sefa( β3x . . ) |   2 |   *   * 192 |  0  0  0  1  1  1 | 0  1  1 1
----------------+-----+-------------+-------------------+----------
both( . x3o . ) |   3 |   3   0   0 | 64  *  *  *  *  * | 1  1  0 0
both( . x . x ) |   4 |   2   2   0 |  * 96  *  *  *  * | 1  0  1 0
both( . . o4x ) |   4 |   0   4   0 |  *  * 48  *  *  * | 1  0  0 1
      β3x . .   ♦   6 |   3   0   3 |  *  *  * 64  *  * | 0  1  1 0
sefa( β3x3o . ) |   3 |   0   0   3 |  *  *  *  * 64  * | 0  1  0 1
sefa( β3x 2 x ) |   4 |   0   2   2 |  *  *  *  *  * 96 | 0  0  1 1
----------------+-----+-------------+-------------------+----------
both( . x3o4x ) ♦  24 |  24  24   0 |  8 12  6  0  0  0 | 8  *  * *
      β3x3o .   ♦  12 |  12   0  12 |  4  0  0  4  4  0 | * 16  * *
      β3x 2 x   ♦  12 |   6   6   6 |  0  3  0  2  0  3 | *  * 32 *
sefa( β3x3o4x ) ♦  24 |   0  24  24 |  0  0  6  0  8 12 | *  *  * 8

starting figure: x3x3o4x

o3x3β4x   (type B)

both( . . . . ) | 192 |   2  1   2  1 |  1  2  2  1  1  2 |  1  1 2  1
----------------+-----+---------------+-------------------+-----------
both( . x . . ) |   2 | 192  *   *  * |  1  1  1  0  0  0 |  1  1 1  0
both( . . . x ) |   2 |   * 96   *  * |  0  2  0  1  0  0 |  1  0 2  0
sefa( . x3β . ) |   2 |   *  * 192  * |  0  0  1  0  1  1 |  0  1 1  1
sefa( . . s4x ) |   2 |   *  *   * 96 |  0  0  0  1  0  2 |  0  0 2  1
----------------+-----+---------------+-------------------+-----------
both( o3x . . ) |   3 |   3  0   0  0 | 64  *  *  *  *  * |  1  1 0  0
both( . x . x ) |   4 |   2  2   0  0 |  * 96  *  *  *  * |  1  0 1  0
      . x3β .   ♦   6 |   3  0   3  0 |  *  * 64  *  *  * |  0  1 1  0
both( . . s4x ) ♦   4 |   0  2   0  2 |  *  *  * 48  *  * |  0  0 2  0
sefa( o3x3β . ) |   3 |   0  0   3  0 |  *  *  *  * 64  * |  0  1 0  1
sefa( . x3β4x ) |   4 |   0  0   2  2 |  *  *  *  *  * 96 |  0  0 1  1
----------------+-----+---------------+-------------------+-----------
both( o3x . x ) ♦   6 |   6  3   0  0 |  2  3  0  0  0  0 | 32  * *  *
      o3x3β .   ♦  12 |  12  0  12  0 |  4  0  4  0  4  0 |  * 16 *  *
      . x3β4x   ♦  48 |  24 24  24 24 |  0 12  8 12  0 12 |  *  * 8  *
sefa( o3x3β4x ) ♦   6 |   0  0   6  3 |  0  0  0  0  2  3 |  *  * * 32

starting figure: o3x3x4x

o3x3β4β   (type B)

both( . . . . ) | 192 |   2   2   2 |  1  2  1  1   4 |  1 2  2
----------------+-----+-------------+-----------------+--------
both( . x . . ) |   2 | 192   *   * |  1  1  0  0   1 |  1 1  1
sefa( . x3β . ) |   2 |   * 192   * |  0  1  0  1   1 |  1 1  1
sefa( . . s4s ) |   2 |   *   * 192 |  0  0  1  0   2 |  0 2  1
----------------+-----+-------------+-----------------+--------
both( o3x . . ) |   3 |   3   0   0 | 64  *  *  *   * |  1 0  1
      . x3β .   ♦   6 |   3   3   0 |  * 64  *  *   * |  1 1  0
both( . . s4s ) ♦   4 |   0   0   4 |  *  * 48  *   * |  0 2  0
sefa( o3x3β . ) |   3 |   0   3   0 |  *  *  * 64   * |  1 0  1
sefa( . x3β4β ) |   4 |   1   1   2 |  *  *  *  * 192 |  0 1  1
----------------+-----+-------------+-----------------+--------
      o3x3β .   ♦  12 |  12  12   0 |  4  4  0  4   0 | 16 *  *
      . x3β4β   ♦  48 |  24  24  48 |  0  8 12  0  24 |  * 8  *
sefa( o3x3β4β ) ♦   6 |   3   3   3 |  1  0  0  1   3 |  * * 64

starting figure: o3x3x4x