  We study the surface states and chiral hinge states of a 3D second-order
topological insulator in the presence of an external magnetic gauge field.
Surfaces pierced by flux host Landau levels, while surfaces parallel to the
applied field are not significantly affected. The chiral hinge modes mediate
spectral flow between neighbouring surfaces. As the magnetic field strength is
increased, the surface Landau quantization deviates from that of a massive
Dirac cone. Quantitatively, the $n = 0$ Landau level falls inside the surface
Dirac gap, and not at the gap edge. The $n \ne 0$ levels exhibit a further,
qualitative discrepancy: while the massive Dirac cone is expected to produce
pairs of levels ($\pm n$) which are symmetric around zero energy, the $n$ and
$-n$ levels become asymmetric in our lattice model -- one of the pair may even
be absent from the spectrum, or hybridized with the continuum. In order to
resolve the issue, we extend the standard 2D massive Dirac surface theory, by
including additional Hamiltonian terms at $\mathcal{O} (k^2)$. While these
terms do not break particle-hole symmetry in the absence of magnetic field,
they lead to the aforementioned Landau level asymmetry once the magnetic field
is applied. We argue that similar $\mathcal{O}(k^2)$ correction terms are
generically expected in lattice models containing gapped Dirac fermions, using
the BHZ model of a 2D topological insulator as an example.
