  We show that the chiral Dirac and Majorana hinge modes in three-dimensional
higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be
gapped while preserving the protecting $\mathsf{C}_{2n}\mathcal T$ symmetry
upon the introduction of non-Abelian surface topological order. In both cases,
the topological order on a single side surface breaks time reversal symmetry,
but appears with its time-reversal conjugate on alternating sides in a
$\mathsf{C}_{2n}\mathcal T$ preserving pattern. In the absence of the
HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on
hinges between $\mathsf{C}_{2n}\mathcal T$-conjugate domains. However, using a
combination of $K$-matrix and anyon condensation arguments, we show that on the
boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence
`anomalous'. Our results suggest that new patterns of surface and hinge states
can be engineered by selectively introducing topological order only on specific
surfaces.
