  Higher-order topological (HOT) phases feature boundary (such as corner and
hinge) modes of codimension $d_c>1$. We here identify an \emph{antiunitary}
operator that ensures the spectral symmetry of a two-dimensional HOT insulator
and the existence of cornered localized states ($d_c=2$) at precise zero
energy. Such an antiunitary symmetry allows us to construct a generalized HOT
insulator that continues to host corner modes even in the presence of a
\emph{weak} anomalous Hall insulator and a spin-orbital density wave orderings,
and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar
conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$
superconductor and the corner localized Majorana zero modes survive even in the
presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also
serve as the building blocks of three-dimensional second-order Weyl semimetals,
supporting one-dimensional hinge modes.
