  The presence or absence of certain symmetries in the normal state (NS) also
determines the symmetry of the Cooper pairs. Here we show that parity
(${\mathcal P}$) and time-reversal (${\mathcal T}$) odd Dirac insulators
(trivial or topological) or metals, sustain a local or intra-unit cell pairing
that supports corner (in $d=2$) or hinge (in $d=3$) modes of Majorana fermions
and stands as a higher-order topological superconductor (HOTSC), when the NS
additionally breaks discrete four-fold ($C_4$) symmetry. Although these
outcomes does not rely on the existence of a Fermi surface, around it (when the
system is doped) the HOTSC takes the form of a mixed parity, ${\mathcal T}$-odd
(due to the lack of ${\mathcal P}$ and ${\mathcal T}$ in the NS, respectively)
$p+id$ pairing, where the $p$($d$)-wave component stems from the Dirac nature
of quasiparticles (lack of $C_4$ symmetry) in the NS. Thus, when strained,
magnetically ordered Dirac materials, such as doped magnetic topological
insulators (MnBi$_2$Te$_4$), can harbor HOTSCs, while the absence of an
external strain should be conducive for the axionic $p+is$ pairing.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons;Superconductivity