  The existing paradigm for topological insulators asserts that an energy gap
separates conduction and valence bands with opposite topological invariants.
Here, we propose that \textit{equal}-energy bands with opposite Chern
invariants can be \textit{spatially} separated -- onto opposite facets of a
finite crystalline Hopf insulator. On a single facet, the number of curvature
quanta is in one-to-one correspondence with the bulk homotopy invariant of the
Hopf insulator -- this originates from a novel bulk-to-boundary flow of Berry
curvature which is \textit{not} a type of Callan-Harvey anomaly inflow. In the
continuum perspective, such nontrivial surface states arise as
\textit{non}-chiral, Schr\"odinger-type modes on the domain wall of a
generalized Weyl equation -- describing a pair of opposite-chirality Weyl
fermions acting as a \textit{dipolar} source of Berry curvature. A
rotation-invariant lattice regularization of the generalized Weyl equation
manifests a generalized Thouless pump -- which translates charge by one lattice
period over \textit{half} an adiabatic cycle, but reverses the charge flow over
the next half.
