  Motivated by recent experiments demonstrating the creation of atomically
sharp interfaces between hexagonal sapphire and cubic SrTiO$_3$ with finite
twist, we here develop and study a general electronic band theory for this
novel class of moir\'e heterostructures. We take into account the
three-dimensional nature of the two crystals, allow for arbitrary combinations
of Bravais lattices, finite twist angles, and different locations in momentum
space of the low-energy electronic bands of the constituent materials. We
analyze the general condition for a well-defined crystalline limit in the
interface electron system and classify the associated "crystalline reference
points". We discuss this in detail for the example of the two-dimensional
lattice planes being square and triangular lattices on the two sides of the
interface; this reveals non-trivial reference points at finite twist angle and
lattice mismatch, leading to a novel form of magic angles, which we refer to as
"geometric magic angles". We further show that band structures of mixed
dimensionality naturally emerge, where quasi-one- and two-dimensional pockets
coexist. Explicit computations for different bulk Bloch Hamiltonians yield a
collection of interesting features, such as isolated bands localized at
interfaces of non-topological insulators, Dirac cones, van Hove singularities,
a non-trivial evolution of the band structures with Zeeman-field, and
topological interface bands. Our work illustrates the potential of these
heterostructures and is anticipated to provide the foundation for moir\'e
interface design and for the analysis of correlated physics in these systems.
