(type A)    (type B)

It looks like a compound of two pentagonal prisms (pip), and indeed vertices, edges, and {4}-faces coincide by pairs.

Incidence matrix according to Dynkin symbol

x x10/2o

. .    . | 20 |  1  2 |  2 1
---------+----+-------+-----
x .    . |  2 | 10  * |  2 0
. x    . |  2 |  * 20 |  1 1
---------+----+-------+-----
x x    . |  4 |  2  2 | 10 *
. x10/2o | 10 |  0 10 |  * 2

x x5/2x

. .   . | 20 |  1  1  1 | 1 1 1
--------+----+----------+------
x .   . |  2 | 10  *  * | 1 1 0
. x   . |  2 |  * 10  * | 1 0 1
. .   x |  2 |  *  * 10 | 0 1 1
--------+----+----------+------
x x   . |  4 |  2  2  0 | 5 * *
x .   x |  4 |  2  0  2 | * 5 *
. x5/2x | 10 |  0  5  5 | * * 2

snubbed forms: s2s5/2s, s2s5/2s (as mere faceting)

x2β5x  

both( . . . ) | 20 |  1  1  1 | 1 1 1
--------------+----+----------+------
both( x . . ) |  2 | 10  *  * | 1 0 1
both( . . x ) |  2 |  * 10  * | 1 1 0
sefa( . β5x ) |  2 |  *  * 10 | 0 1 1
--------------+----+----------+------
both( x . x ) |  4 |  2  2  0 | 5 * *
      . β5x   ♦ 10 |  0  5  5 | * 2 *
sefa( x2β5x ) |  4 |  2  0  2 | * * 5

starting figure: x x5x

β2β5x  

both( . . . ) | 20 |  1  1  1 | 1  2
--------------+----+----------+-----
both( . . x ) |  2 | 10  *  * | 1  1
both( s2s . ) |  2 |  * 10  * | 0 2
sefa( . β5x ) |  2 |  *  * 10 | 1  1
--------------+----+----------+-----
      . β5x   ♦ 10 |  5  0  5 | 2 *
sefa( β2β5x ) |  4 |  1  2  1 | * 10

starting figure: x x5x

xx10/2oo&#x   → height = 1
({10/2} || {10/2})

o.10/2o.    | 10  * |  2  1  0 | 1  2 0
.o10/2.o    |  * 10 |  0  1  2 | 0  2 1
------------+-------+----------+-------
x.    ..    |  2  0 | 10  *  * | 1  1 0
oo10/2oo&#x |  1  1 |  * 10  * | 0  2 0
.x    ..    |  0  2 |  *  * 10 | 0  1 1
------------+-------+----------+-------
x.10/2o.    | 10  0 | 10  0  0 | 1  * *
xx    ..&#x |  2  2 |  1  2  1 | * 10 *
.x10/2.o    |  0 10 |  0  0 10 | *  * 1

xx5/2xx&#x   → height = 1
({10/2} || {10/2})

o.5/2o.    | 10  * | 1 1  1 0 0 | 1 1 1 0
.o5/2.o    |  * 10 | 0 0  1 1 1 | 0 1 1 1
-----------+-------+------------+--------
x.   ..    |  2  0 | 5 *  * * * | 1 1 0 0
..   x.    |  2  0 | * 5  * * * | 1 0 1 0
oo5/2oo&#x |  1  1 | * * 10 * * | 0 1 1 0
.x   ..    |  0  2 | * *  * 5 * | 0 1 0 1
..   .x    |  0  2 | * *  * * 5 | 0 0 1 1
-----------+-------+------------+--------
x.5/2x.    | 10  0 | 5 5  0 0 0 | 1 * * *
xx   ..&#x |  2  2 | 1 0  2 1 0 | * 5 * *
..   xx&#x |  2  2 | 0 1  2 0 1 | * * 5 *
.x5/2.x    |  0 10 | 0 0  0 5 5 | * * * 1