(type A)    (type B)

Looks like a compound of 2 coincident n-gonal prisms (n-p), and indeed all elements (but the Grünbaumian bases) coincide by pairs.

Incidence matrix according to Dynkin symbol

x xn/2x   (n>2)

. .   . | 4n |  1  1  1 | 1 1 1
--------+----+----------+------
x .   . |  2 | 2n  *  * | 1 1 0
. x   . |  2 |  * 2n  * | 1 0 1
. .   x |  2 |  *  * 2n | 0 1 1
--------+----+----------+------
x x   . |  4 |  2  2  0 | n * *
x .   x |  4 |  2  0  2 | * n *
. xn/2x | 2n |  0  n  n | * * 2

x2βnx   (n>2)

both( . . . ) | 4n |  1  1  1 | 1 1 1
--------------+----+----------+------
both( x . . ) |  2 | 2n  *  * | 0 1 1
both( . . x ) |  2 |  * 2n  * | 1 1 0
sefa( . βnx ) |  2 |  *  * 2n | 1 0 1
--------------+----+----------+------
      . βnx   ♦ 2n |  0  n  n | 2 * *
both( x . x ) |  4 |  2  2  0 | * n *
sefa( x2βnx ) |  4 |  2  0  2 | * * n

starting figure: x xnx

β2βnx   (n>2)

both( . . . ) | 4n |  1  1  1 | 1  2
--------------+----+----------+-----
both( s2s   ) |  2 | 2n  *  * | 0  2
both( . . x ) |  2 |  * 2n  * | 1  1
sefa( . βnx ) |  2 |  *  * 2n | 1  1
--------------+----+----------+-----
        βnx   ♦ 2n |  0  n  n | 2  *
sefa( β2βnx ) |  4 |  2  1  1 | * 2n

starting figure: x xnx

xxn/2xx&#x   (n>2)   → height = 1
({2n/2} || {2n/2})

o.n/2o.    | 2n  * | 1 1  1 0 0 | 1 1 1 0
.on/2.o    |  * 2n | 0 0  1 1 1 | 0 1 1 1
-----------+-------+------------+--------
x.   ..    |  2  0 | n *  * * * | 1 1 0 0
..   x.    |  2  0 | * n  * * * | 1 0 1 0
oon/2oo&#x |  1  1 | * * 2n * * | 0 1 1 0
.x   ..    |  0  2 | * *  * n * | 0 1 0 1
..   .x    |  0  2 | * *  * * n | 0 0 1 1
-----------+-------+------------+--------
x.n/2x.    | 2n  0 | n n  0 0 0 | 1 * * *
xx   ..&#x |  2  2 | 1 0  2 1 0 | * n * *
..   xx&#x |  2  2 | 0 1  2 0 1 | * * n *
.xn/2.x    |  0 2n | 0 0  0 n n | * * * 1