  Planar bipartite graphs can be represented as touching graphs of horizontal
and vertical segments in $\mathbb{R}^2$. We study a generalization in space:
touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that
planar 3-colorable graphs can be represented this way. The result implies a
characterization of corner polytopes previously obtained by Eppstein and
Mumford. A by-product of our proof is a distributive lattice structure on the
set of orthogonal surfaces with given skeleton. Further, we study
representations by axis-aligned non-coplanar rectangles in $\mathbb{R}^3$ such
that all regions are boxes. We show that the resulting graphs correspond to
octahedrations of an octahedron. This generalizes the correspondence between
planar quadrangulations and families of horizontal and vertical segments in
$\mathbb{R}^2$ with the property that all regions are rectangles.
