Looks like a compound of 2 octahedra (oct) plus the 3 pairs of diametral squares each, and indeed edges and {4} coincide by four, but vertices and {3} coincide by pairs. Just as oct+6{4} also can be seen as joined pair of mutually inverted thah, this one can be seen as joined pair of mutually inverted 2thah.

Incidence matrix according to Dynkin symbol

β3/2β3o3*a

both( .   . .    ) | 12 |  4  2  2 | 2 1 1  4
-------------------+----+----------+---------
sefa( s3/2s .    ) |  2 | 24  *  * | 1 0 0  1
sefa( β   . o3*a ) |  2 |  * 12  * | 0 1 0  1
sefa( .   β3o    ) |  2 |  *  * 12 | 0 0 1  1
-------------------+----+----------+---------
both( s3/2s .    ) ♦  3 |  3  0  0 | 8 * *  *
      β   . o3*a   ♦  3 |  0  3  0 | * 4 *  *
      .   β3o      ♦  3 |  0  0  3 | * * 4  *
sefa( β3/2β3o3*a ) |  4 |  2  1  1 | * * * 12

starting figure: x3/2x3o3*a

β3/2β3/2o3/2*a

both( .   .   .      ) | 12 |  4  2  2 | 2 1 1  4
-----------------------+----+----------+---------
sefa( s3/2s   .      ) |  2 | 24  *  * | 1 0 0  1
sefa( β   .   o3/2*a ) |  2 |  * 12  * | 0 1 0  1
sefa( .   β3/2o      ) |  2 |  *  * 12 | 0 0 1  1
-----------------------+----+----------+---------
both( s3/2s   .      ) ♦  3 |  3  0  0 | 8 * *  *
      β   .   o3/2*a   ♦  3 |  0  3  0 | * 4 *  *
      .   β3/2o        ♦  3 |  0  0  3 | * * 4  *
sefa( β3/2β3/2o3/2*a ) |  4 |  2  1  1 | * * * 12

starting figure: x3/2x3/2o3/2*a

β3/2o3β3*a

both( .   . .    ) | 12 |  2  4  2 | 1 2 1  4
-------------------+----+----------+---------
sefa( β3/2o .    ) |  2 | 12  *  * | 1 0 0  1
sefa( s   . s3*a ) |  2 |  * 24  * | 0 1 0  1
sefa( .   o3β    ) |  2 |  *  * 12 | 0 0 1  1
-------------------+----+----------+---------
      β3/2o .      ♦  3 |  3  0  0 | 8 * *  *
both( s   . s3*a ) ♦  3 |  0  3  0 | * 4 *  *
      .   o3β      ♦  3 |  0  0  3 | * * 4  *
sefa( β3/2o3β3*a ) |  4 |  1  2  1 | * * * 12

starting figure: x3/2o3x3*a