  One of the hallmarks of topological insulators is the correspondence between
the value of its bulk topological invariant and the number of topologically
protected edge modes observed in a finite-sized sample. This bulk-boundary
correspondence has been well-tested for strong topological invariants, and
forms the basis for all proposed technological applications of topology. Here,
we report that a group of weak topological invariants, which depend only on the
symmetries of the atomic lattice, also induces a particular type of
bulk-boundary correspondence. It predicts the presence or absence of states
localised at the interface between two inversion-symmetric band insulators with
trivial values for their strong invariants, based on the space group
representation of the bands on either side of the junction. We show that this
corresponds with symmetry-based classifications of topological materials. The
interface modes are protected by the combination of band topology and symmetry
of the interface, and may be used for topological transport and signal
manipulation in heterojunction-based devices.
