Looks like a compound of 2 gishi with vertices and edges identified, while faces and cells coincide by pairs.

As abstract polytope 2gishi is isomorphic to 2gahi, thereby replacing gissid by doe, resp. replacing pentagrammal faces by pentagonal ones, resp. replacing 2ike vertex figures by 2gike ones.

Incidence matrix according to Dynkin symbol

x5/2o3o5/2o3*b

.   . .   .    | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/2o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/2o3o   .    ♦  20 |  30 |   12 | 120   *
x5/2o .   o3*b ♦  20 |  30 |   12 |   * 120

or
.   . .   .       | 120 ♦  12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/2o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/2o3o   .     & ♦  20 |  30 |   12 | 240

x5/2o3o5/3o3/2*b

.   . .   .      | 120 ♦  12 |   60 |  20  20
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/2o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/2o3o   .      ♦  20 |  30 |   12 | 120   *
x5/2o .   o3/2*b ♦  20 |  30 |   12 |   * 120

x5/2o3/2o5/2o3/2*b

.   .   .   .      | 120 ♦  12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/2o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/2o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/2o   .   o3/2*b ♦  20 |  30 |   12 |   * 120

or
.   .   .   .         | 120 ♦  12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/2o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/2o3/2o   .       & ♦  20 |  30 |   12 | 240

x5/3o3o5/2o3*b

.   . .   .    | 120 ♦  12 |   60 |  20  20
---------------+-----+-----+------+--------
x   . .   .    |   2 | 720 |   10 |   5   5
---------------+-----+-----+------+--------
x5/3o .   .    |   5 |   5 | 1440 |   1   1
---------------+-----+-----+------+--------
x5/3o3o   .    ♦  20 |  30 |   12 | 120   *
x5/3o .   o3*b ♦  20 |  30 |   12 |   * 120

or
.   . .   .       | 120 ♦  12 |   60 |  40
------------------+-----+-----+------+----
x   . .   .       |   2 | 720 |   10 |  10
------------------+-----+-----+------+----
x5/3o .   .       |   5 |   5 | 1440 |   2
------------------+-----+-----+------+----
x5/3o3o   .     & ♦  20 |  30 |   12 | 240

x5/3o3o5/3o3/2*b

.   . .   .      | 120 ♦  12 |   60 |  20  20
-----------------+-----+-----+------+--------
x   . .   .      |   2 | 720 |   10 |   5   5
-----------------+-----+-----+------+--------
x5/3o .   .      |   5 |   5 | 1440 |   1   1
-----------------+-----+-----+------+--------
x5/3o3o   .      ♦  20 |  30 |   12 | 120   *
x5/3o .   o3/2*b ♦  20 |  30 |   12 |   * 120

x5/3o3/2o5/2o3/2*b

.   .   .   .      | 120 ♦  12 |   60 |  20  20
-------------------+-----+-----+------+--------
x   .   .   .      |   2 | 720 |   10 |   5   5
-------------------+-----+-----+------+--------
x5/3o   .   .      |   5 |   5 | 1440 |   1   1
-------------------+-----+-----+------+--------
x5/3o3/2o   .      ♦  20 |  30 |   12 | 120   *
x5/3o   .   o3/2*b ♦  20 |  30 |   12 |   * 120

or
.   .   .   .         | 120 ♦  12 |   60 |  40
----------------------+-----+-----+------+----
x   .   .   .         |   2 | 720 |   10 |  10
----------------------+-----+-----+------+----
x5/3o   .   .         |   5 |   5 | 1440 |   2
----------------------+-----+-----+------+----
x5/3o3/2o   .       & ♦  20 |  30 |   12 | 240