(type A)    (type B)

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

Incidence matrix according to Dynkin symbol

x x6/2o

. .   . | 12 | 1  2 | 2 1
--------+----+------+----
x .   . |  2 | 6  * | 2 0
. x   . |  2 | * 12 | 1 1
--------+----+------+----
x x   . |  4 | 2  2 | 6 *
. x6/2o |  6 | 0  6 | * 2

snubbed forms: s2s6/2o

x x3/2x

. .   . | 12 | 1 1 1 | 1 1 1
--------+----+-------+------
x .   . |  2 | 6 * * | 1 1 0
. x   . |  2 | * 6 * | 1 0 1
. .   x |  2 | * * 6 | 0 1 1
--------+----+-------+------
x x   . |  4 | 2 2 0 | 3 * *
x .   x |  4 | 2 0 2 | * 3 *
. x3/2x |  6 | 0 3 3 | * * 2

snubbed forms: s2s3/2s

x2β3x   (type A)

both( . . . ) | 12 | 1 1 1 | 1 1 1
--------------+----+-------+------
both( x . . ) |  2 | 6 * * | 1 0 1
both( . . x ) |  2 | * 6 * | 1 1 0
sefa( . β3x ) |  2 | * * 6 | 0 1 1
--------------+----+-------+------
both( x . x ) |  4 | 2 2 0 | 3 * *
      . β3x   ♦  6 | 0 3 3 | * 2 *
sefa( x2β3x ) |  4 | 2 0 2 | * * 3

starting figure: x x3x

β2β3x   (type B)

both( . . . ) | 12 | 1 1 1 | 1 2
--------------+----+-------+----
both( . . x ) |  2 | 6 * * | 1 1
both( s2s . ) |  2 | * 6 * | 0 2
sefa( . β3x ) |  2 | * * 6 | 1 1
--------------+----+-------+----
      . β3x   ♦  6 | 3 0 3 | 2 *
sefa( β2β3x ) |  4 | 1 2 1 | * 6

starting figure: x x3x

xx6/2oo&#x   → height = 1
({6/2} || {6/2})

o.6/2o.    | 6 * | 2 1 0 | 1 2 0
.o6/2.o    | * 6 | 0 1 2 | 0 2 1
-----------+-----+-------+------
x.   ..    | 2 0 | 6 * * | 1 1 0
oo6/2oo&#x | 1 1 | * 6 * | 0 2 0
.x   ..    | 0 2 | * * 6 | 0 1 1
-----------+-----+-------+------
x.6/2o.    | 6 0 | 6 0 0 | 1 * *
xx   ..&#x | 2 2 | 1 2 1 | * 6 *
.x6/2.o    | 0 6 | 0 0 6 | * * 1

snubbed forms: ss6/2oo&#x

xx3/2xx&#x   → height = 1
({6/2} || {6/2})

o.3/2o.    | 6 * | 1 1 1 0 0 | 1 1 1 0
.o3/2.o    | * 6 | 0 0 1 1 1 | 0 1 1 1
-----------+-----+-----------+--------
x.   ..    | 2 0 | 3 * * * * | 1 1 0 0
..   x.    | 2 0 | * 3 * * * | 1 0 1 0
oo3/2oo&#x | 1 1 | * * 6 * * | 0 1 1 0
.x   ..    | 0 2 | * * * 3 * | 0 1 0 1
..   .x    | 0 2 | * * * * 3 | 0 0 1 1
-----------+-----+-----------+--------
x.3/2x.    | 6 0 | 3 3 0 0 0 | 1 * * *
xx   ..&#x | 2 2 | 1 0 2 1 0 | * 3 * *
..   xx&#x | 2 2 | 0 1 2 0 1 | * * 3 *
.x3/2.x    | 0 6 | 0 0 0 3 3 | * * * 1

snubbed forms: ss3/2ss&#x