  Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian
topological phases are shown to support a remarkably rich sequence of 1D
symmetry-protected topological (SPT) states. We show that such interfaces can
provide a physical interpretation for the corrections to the topological
entanglement entropy of a 2D state with Abelian topological order found in
Ref.~\onlinecite{cano-2015}. The topological entanglement entropy decomposes as
$\gamma = \gamma_a + \gamma_s$, where $\gamma_a > 0$ only depends on universal
topological properties of the 2D state, while a correction $\gamma_s > 0$
signals the emergence of the 1D SPT state that is produced by interactions
along the entanglement cut and provides a direct measure of the stabilizing
symmetry of the resulting SPT state. A correspondence is established between
the possible values of $\gamma_s$ associated with a given interface - which is
named Boundary Topological Entanglement Sequence - and classes of 1D SPT
states. We show that symmetry-preserving domain walls along such 1D interfaces
(or boundaries) generally host localized parafermion-like excitations that are
stable to local symmetry-preserving perturbations.
