Both the taller and shallower gissid || sissid have the same incidence matrix. In fact the bases are aligned in the same mutual orientation, just within different heights.

Within the shallower one the starps connect anti-aligned pentagrams on the same side each, while in the taller one the staps connect aligned pentagrams on the opposite side each. Likewise, in the shallower one tets connect orthogonal edges on the opposite side each, while in the taller one the tets connect orthogonal edges on the same side each.

Similarily in the shallower one the axially trigonal vertex figure at the gissid base consists of 2 aligned triangles (of different sizes) and the lacing tetragons are crossed trapezoids, while in the taller one the axially trigonal vertex figure at the gissid base consists of 2 anti-aligned triangles (of different sizes) and the lacing tetragons are convex trapezoids. Alike in the shallower one the axially pentagonal vertex figure at the sissid base is a segmentohedron spanned by an anti-aligned pentagon-pentagram pair (of same circumradius) and the lacing tetragons are crossed trapezoids, while in the taller one the axially pentagonal vertex figure at the sissid base is a segmentohedron spanned by an aligned pentagon-pentagram pair (of same circumradius) and the lacing tetragons are convex trapezoids.

Incidence matrix

gissid || sissid   → height = sqrt[sqrt(5)-7/4] = 0.697186

20  * |  3  3  0 |  3  6  3  0 | 1  3  3 0
 * 12 |  0  5  5 |  0  5 10  5 | 0  5  5 1
------+----------+-------------+----------
 2  0 | 30  *  * |  2  2  0  0 | 1  2  1 0
 1  1 |  * 60  * |  0  2  2  0 | 0  2  2 0
 0  2 |  *  * 30 |  0  0  2  2 | 0  2  1 1
------+----------+-------------+----------
 5  0 |  5  0  0 | 12  *  *  * | 1  1  0 0  {5/2}
 2  1 |  1  2  0 |  * 60  *  * | 0  1  1 0
 1  2 |  0  2  1 |  *  * 60  * | 0  1  1 0
 0  5 |  0  0  5 |  *  *  * 12 | 0  1  0 1  {5/2}
------+----------+-------------+----------
20  0 | 30  0  0 | 12  0  0  0 | 1  *  * * gissid
 5  5 |  5 10  5 |  1  5  5  1 | * 12  * * stap
 2  2 |  1  4  1 |  0  2  2  0 | *  * 30 * tet
 0 12 |  0  0 30 |  0  0  0 12 | *  *  * 1 sissid