We can generate embedded polyhedra by a simple technique.
Between any two vertices which make an edge, we create a new vertex
for the embedded polyhedra in the middle of that edge. Joining
those new vertices generates the new solid. For the case of the
cube, this process immediately results in a truncated cube, the
cuboctahedron. Curiously enough, we can also generate the
cuboctahedron by two stages of dividing the tetrahedron. The end of the first
stage results in an octohedron. Dividing the octahedron
results in a cuboctahedron.

Geomags are perfect for building these geometrical objects.

In the rendered image, I've colored the three planes of the
internal octohedron red, green and blue.

Here is the isolated octohedron. Again, notice the orthogonal
bisecting planes.

Now we take the octohedron, magnify by two, and create nodes at
the edge midpoints. Eliminate the original nodes, and use the
central node for bracing gives us a rigid cuboctahedron.

While the tetrahedron and octohedron are stiff structures, the
cube and cuboctahedron are not. Internally bracing the
cuboctahedron with a twelve point star does achieve a stiff
structure with cartesian orientation and stackability.

Rendered images, of course, don't care about mechanical stability.
Here is the cuboctahedron rendered, with four bisecting planes with
hexagon cross sections, colored red, green, blue and yellowish.