  We introduce novel higher-order topological phases of matter in
chiral-symmetric systems (class AIII of the tenfold classification), most of
which would be misidentified as trivial by current theories. These phases are
protected by "multipole chiral numbers", bulk integer topological invariants
that in 2D and 3D are built from sublattice multipole moment operators, as
defined herein. The integer value of a multipole chiral number indicates how
many degenerate zero-energy states localize at each corner of a system. These
higher-order phases of matter are generally boundary-obstructed and robust in
the presence of chiral-symmetry-preserving disorder.
