  Dirac semimetals lack a simple bulk-boundary correspondence. Recently, Dirac
materials with four-fold rotation symmetry have been shown to exhibit a higher
order bulk-hinge correspondence: they display "higher order Fermi arcs," which
are localized on hinges where two surfaces meet and connect the projections of
the bulk Dirac points. In this paper, we classify higher order Fermi arcs for
Dirac semimetals protected by a rotation symmetry and the product of
time-reversal and inversion. Such Dirac points can be either linear in all
directions or linear along the rotation axis and quadratic in other directions.
By computing the filling anomaly for momentum-space planes on either side of
the Dirac point, we find that all linear Dirac points exhibit higher order
Fermi arcs terminating at the projection of the Dirac point, while the Dirac
points that are quadratic in two directions lack such higher order Fermi arcs.
When higher order Fermi arcs do exist, they obey either a $\mathbb{Z}_2$
(four-fold rotation axis) or $\mathbb{Z}_3$ (three- or six-fold rotation axis)
group structure. Finally, we build two models with six-fold symmetry to
illustrate the cases with and without higher order Fermi arcs. We predict
higher order Fermi arcs in Na$_3$Bi.
