Looks like the compound of 2 prismatorhombated icositetrachora (prico), and indeed all but the Grünbaumian elements coincide by pairs.

Incidence matrix according to Dynkin symbol

x3β4x3x

both( . . . . ) | 1152 |   1   1   1   1   1 |   1   1   1   1   1   1   1   1 |  1  1  1  1  1
----------------+------+---------------------+---------------------------------+---------------
both( x . . . ) |    2 | 576   *   *   *   * |   1   1   0   1   0   0   0   0 |  1  1  1  0  0
both( . . x . ) |    2 |   * 576   *   *   * |   1   0   1   0   1   0   0   0 |  1  1  0  1  0
both( . . . x ) |    2 |   *   * 576   *   * |   0   1   1   0   0   0   1   1 |  1  0  1  1  1
sefa( x3β . . ) |    2 |   *   *   * 576   * |   0   0   0   1   0   1   1   0 |  0  1  1  0  1
sefa( . s4x . ) |    2 |   *   *   *   * 576 |   0   0   0   0   1   1   0   1 |  0  1  0  1  1
----------------+------+---------------------+---------------------------------+---------------
both( x . x . ) |    4 |   2   2   0   0   0 | 288   *   *   *   *   *   *   * |  1  1  0  0  0
both( x . . x ) |    4 |   2   0   2   0   0 |   * 288   *   *   *   *   *   * |  1  0  1  0  0
both( . . x3x ) |    6 |   0   3   3   0   0 |   *   * 192   *   *   *   *   * |  1  0  0  1  0
      x3β . .   ♦    6 |   3   0   0   3   0 |   *   *   * 192   *   *   *   * |  0  1  1  0  0
both( . s4x . ) ♦    4 |   0   2   0   0   2 |   *   *   *   * 288   *   *   * |  0  1  0  1  0
sefa( x3β4x . ) |    4 |   0   0   0   2   2 |   *   *   *   *   * 288   *   * |  0  1  0  0  1
sefa( x3β 2 x ) |    4 |   0   0   2   2   0 |   *   *   *   *   *   * 288   * |  0  0  1  0  1
sefa( . s4x3x ) |    6 |   0   0   3   0   3 |   *   *   *   *   *   *   * 192 |  0  0  0  1  1
----------------+------+---------------------+---------------------------------+---------------
both( x . x3x ) ♦   12 |   6   6   6   0   0 |   3   3   2   0   0   0   0   0 | 96  *  *  *  *
      x3β4x .   ♦   48 |  24  24   0  24  24 |  12   0   0   8  12  12   0   0 |  * 24  *  *  *
      x3β 2 x   ♦   12 |   6   0   6   6   0 |   0   3   0   2   0   0   3   0 |  *  * 96  *  *
both( . s4x3x ) ♦   24 |   0  12  12   0  12 |   0   0   4   0   6   0   0   4 |  *  *  * 48  *
sefa( x3β4x3x ) ♦   12 |   0   0   6   6   6 |   0   0   0   0   0   3   3   2 |  *  *  *  * 96

starting figure: x3x4x3x