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This toroidal polyhedron is the blend of id with 2 thawroes. In fact, the latter uses parts of the former's boundary and additionally has exactly half its height, cf. the right pic.

Incidence matrix according to Dynkin symbol

fofFxx3xFfofx&#xr 
(axially tunneled toroid)

o.....3o.....     & | 12 * * * | 1  1  1  1  0  0 0 |  1 1  1 1 0
.o....3.o....     & |  * 6 * * | 0  2  0  0  2  2 0 |  2 1  2 0 1
..o...3..o...       |  * * 6 * | 0  0  2  0  2  0 0 |  2 2  0 0 0
.....o3.....o       |  * * * 6 | 0  0  0  2  0  2 2 |  0 0  2 2 2
--------------------+----------+--------------------+------------
...... x.....     & |  2 0 0 0 | 6  *  *  *  *  * * |  0 1  0 1 0
oo....3oo....&#x  & |  1 1 0 0 | * 12  *  *  *  * * |  1 0  1 0 0
o.o...3o.o...&#x  & |  1 0 1 0 | *  * 12  *  *  * * |  1 1  0 0 0
o....o3o....o&#x  & |  1 0 0 1 | *  *  * 12  *  * * |  0 0  1 1 0
.oo...3.oo...&#x  & |  0 1 1 0 | *  *  *  * 12  * * |  1 1  0 0 0
.o...o3.o...o&#x  & |  0 1 0 1 | *  *  *  *  * 12 * |  0 0  1 0 1
.....x ......     & |  0 0 0 2 | *  *  *  *  *  * 6 |  0 0  0 1 1
--------------------+----------+--------------------+------------
ooo...3ooo...&#x  & |  1 1 1 0 | 0  1  1  0  1  0 0 | 12 *  * * *
...... x.fo..&#xt & |  2 1 2 0 | 1  0  2  0  2  0 0 |  * 6  * * *  {5}
oo...o3oo...o&#x  & |  1 1 0 1 | 0  1  0  1  0  1 0 |  * * 12 * *
...... x....x&#x  & |  2 0 0 2 | 1  0  0  2  0  0 1 |  * *  * 6 *  {4}
.o...x ......&#x  & |  0 1 0 2 | 0  0  0  0  0  2 1 |  * *  * * 6