  Orthogonal surfaces are nice mathematical objects which have interesting
connections to various fields, e.g., integer programming, monomial ideals and
order dimension. While orthogonal surfaces in one or two dimensions are rather
trivial already the three dimensional case has a rich structure with
connections to Schnyder woods, planar graphs and 3-polytopes.
  Our objective is to detect more of the structure of orthogonal surfaces in
four and higher dimensions. In particular we are driven by the question which
non-generic orthogonal surfaces have a polytopal structure.
  We study characteristic points and the cp-orders of orthogonal surfaces,
i.e., the dominance orders on the characteristic points. In the generic case
these orders are (almost) face lattices of polytopes. Examples show that in
general cp-orders can lack key properties of face lattices. We investigate
extra requirements which may help to have cp-orders which are face lattices.
  Finally, we turn the focus and ask for the realizability of polytopes on
orthogonal surfaces. There are criteria which prevent large classes of
simplicial polytopes from being realizable. On the other hand we identify some
families of polytopes which can be realized on orthogonal surfaces.
