Looks like a compound of 2 tetrahedra (tet), in type A and B with vertices identified, edges and faces both coincide by pairs. While in type A there indeed exist paths (when not crossing vertices) onto all the 8 faces, in type B they remain separate subsets of 4 each. Thence type B more can be viewed as a the mentioned coincident compound itself with just the vertices being identified.

Type C more can be considered as a {6/2} pyramid instead, i.e. the tip is a single vertex, the others come in coincident pairs and all the edges come in coincident pairs too. Having 2 different vertex types C will not be uniform. But that one will be self-dual instead.

And then there is surely type D, defined as the dual of type A, as well. That one then clearly is not uniform either, as it has not identified coincident vertices and pairwise coincident edges too, but at least it will be isohedral again.

Incidence matrix according to Dynkin symbol

x3/2o3o3*a   (type A)

.   . .    | 4 |  6 | 3 3
-----------+---+----+----
x   . .    | 2 | 12 | 1 1
-----------+---+----+----
x3/2o .    | 3 |  3 | 4 *
x   . o3*a | 3 |  3 | * 4

snubbed forms: β3/2o3o3*a

o3/2o3x3*a   (type A)

.   . .    | 4 |  6 | 3 3
-----------+---+----+----
.   . x    | 2 | 12 | 1 1
-----------+---+----+----
.   o3x    | 3 |  3 | 4 *
o   . x3*a | 3 |  3 | * 4

snubbed forms: o3/2o3β3*a

x3/2o3/2o3/2*a   (type A)

.   .   .      | 4 |  6 | 3 3
---------------+---+----+----
x   .   .      | 2 | 12 | 1 1
---------------+---+----+----
x3/2o   .      | 3 |  3 | 4 *
x   .   o3/2*a | 3 |  3 | * 4

snubbed forms: β3/2o3/2o3/2*a

β3o3o   (type A)

both( . . . ) | 4 |  6 | 3 3
--------------+---+----+----
sefa( β3o . ) | 2 | 12 | 1 1
--------------+---+----+----
      β3o .   ♦ 3 |  3 | 4 *
sefa( β3o3o ) | 3 |  3 | * 4

starting figure: x3o3o

β3/2o3o   (type A)

both( .   . . ) | 4 |  6 | 3 3
----------------+---+----+----
sefa( β3/2o . ) | 2 | 12 | 1 1
----------------+---+----+----
      β3/2o .   ♦ 3 |  3 | 4 *
sefa( β3/2o3o ) | 3 |  3 | * 4

starting figure: x3/2o3o

o3/2o3β   (type A)

both( .   . . ) | 4 |  6 | 3 3
----------------+---+----+----
sefa( .   o3β ) | 2 | 12 | 1 1
----------------+---+----+----
      .   o3β   ♦ 3 |  3 | 4 *
sefa( o3/2o3β ) | 3 |  3 | * 4

starting figure: o3/2o3x

β3/2o3/2o   (type A)

both( .   .   . ) | 4 |  6 | 3 3
------------------+---+----+----
sefa( β3/2o   . ) | 2 | 12 | 1 1
------------------+---+----+----
      β3/2o   .   ♦ 3 |  3 | 4 *
sefa( β3/2o3/2o ) | 3 |  3 | * 4

starting figure: x3/2o3/2o

β3/2x3o   (type A)

third( .   . . ) | 4 |  6 | 3 3
-----------------+---+----+----
both(  .   x . ) | 2 | 12 | 1 1
-----------------+---+----+----
both(  .   x3o ) | 3 |  3 | 4 *
       β3/2x .   ♦ 3 |  3 | * 4

starting figure: x3/2x3o

(Here sefa( β3/2x3o ) degenerates to a point,
thus 3 points each from x3/2x3o will be identified.)

x3/2β3o   (type B)

third( .   . . ) | 4 | 3 3 | 3 3
-----------------+---+-----+----
both(  x   . . ) | 2 | 6 * | 2 0
sefa(  .   β3o ) | 2 | * 6 | 0 2
-----------------+---+-----+----
       x3/2β .   ♦ 3 | 3 0 | 4 *
       .   β3o   ♦ 3 | 0 3 | * 4

starting figure: x3/2x3o
       
(Here the tetragons sefa( x3/2β3o ) degenerate to a line,
thus their remaining edges, sefa( β3o ), will be identified,
alike pairwise their vertices.)

ox6/2oo&#x   (type C)   → height = sqrt(2/3) = 0.816497

o.6/2o.    | 1 * | 6 0 | 6 0  [36]/2
.o6/2.o    | * 6 | 1 2 | 2 1  [32,6/2]
-----------+-----+-----+----
oo6/2oo&#x | 1 1 | 6 * | 2 0  come in coincident pairs
.x   ..    | 0 2 | * 6 | 1 1  come in coincident pairs
-----------+-----+-----+----
ox   ..&#x | 1 2 | 2 1 | 6 *  {3}
.x6/2.o    | 0 6 | 0 6 | * 1  {6/2}

ox3/2ox&#x   (type C)   → height = sqrt(2/3) = 0.816497

o.3/2o.    | 1 * | 6 0 0 | 3 3 0  [36]/2
.o3/2.o    | * 6 | 1 1 1 | 1 1 1  [32,6/2]
-----------+-----+-------+------
oo3/2oo&#x | 1 1 | 6 * * | 1 1 0  come in coincident pairs
.x   ..    | 0 2 | * 6 * | 1 0 1  coincident with the next
..   .x    | 0 2 | * 6*  | 0 1 1  coincident with the former
-----------+-----+-------+------
ox   ..&#x | 1 2 | 2 1 0 | 3 * *  {3}
..   ox&#x | 1 2 | 2 0 1 | * 3 *  {3}
.x3/2.x    | 0 6 | 0 3 3 | * * 1  {6/2}

(type D)

4 * |  3 | 3  [(6/2)3] coincident with the next
* 4 |  3 | 3  [(6/2)3] coincident with the former
----+----+--
1 1 | 12 | 2  come in coincident pairs
----+----+--
3 3 |  6 | 4  {6/2}