Pairs of mutually complementary $2$-dimensional simplicial polyhedra: Interesting examples

We construct an example of a pair of ($2$-dimensional) $8$-vertex simplicial toroidal polyhedra (each polyhedron without self-intersection) with same $1$-dimensional skeleton in (Euclidean) $3$-space, which do not have a single common $2$-face, and the union of the $2$-skeletons of these two polyhedra gives a geometric realization of the $2$-skeleton of the $4$-dimensional hyperoctahedron in $3$-space. Also, we construct an example of a pair of $6$-vertex simplicial polyhedral projective planes with the same $1$-skeleton in $4$-space, which do not have a single common $2$-face, and the union of these projective planes gives a geometric realization of the $2$-skeleton of the $5$-hypertetrahedron in $4$-space. Finally, it is shown how to imagine, figuratively, the atoms in the molecule of methane ${\rm{CH}}_4$ “linked” by a pair of internally disjoint spanning polyhedral Möbius strips.