  Higher-order phases are characterized by corner or hinge modes that arise due
to the interesting interplay of localization mechanisms along two or more
dimensions. In this work, we introduce and construct a novel class of "hybrid"
higher-order skin-topological boundary modes in non-reciprocal systems with two
or more open boundaries. Their existence crucially relies on non-reciprocal
pumping in addition to topological localization. Unlike usual non-Hermitian
"skin" modes, they can exist in lattices with \emph{vanishing} net reciprocity
due to the selective nature of non-reciprocal pumping: While the bulk modes
remain extended due to the cancellation of non-reciprocity within each unit
cell, boundary modes experience a curious \emph{spontaneous breaking} of
reciprocity in the presence of topological localization, thereby experiencing
the non-Hermitian skin effect. The number of possible hybridization channels
increases rapidly with dimensionality, leading to a proliferation of distinct
phases. In addition, skin modes or hybrid skin-topological modes can restore
unitarity and are hence stable, allowing for experimental observations and
manipulations in non-Hermitian photonic and electrical metamaterials.
