  The quantum anomalous Hall (QAH) effect, a condensed matter analog of the
parity anomaly, is characterized by a quantized Hall conductivity in the
absence of an external magnetic field. However, it has been recently shown
that, even in the presence of Landau levels, the QAH effect can be
distinguished from the conventional quantum Hall (QH) effect due to the parity
anomaly. As a signature of this effect, we predicted coexistent
counterpropagating QAH and QH edge states. In the present work, we generalize
these findings to QAH insulators with broken particle-hole symmetry. In
particular, we derive the connection to the spectral asymmetry, which is a
topological quantity arising in the context of Dirac-like systems. Moreover, it
is shown that, depending on the magnetic field direction, particle-hole
asymmetry strengthens or weakens the hybridization of the counterpropagating QH
and QAH edge states. Implications for ferro- or paramagnetic topological
insulators are derived which paves the way towards identifying signatures of
the QAH effect even in external magnetic fields.
