  Josephson junctions based on three-dimensional topological insulators offer
intriguing possibilities to realize unconventional $p$-wave pairing and
Majorana modes. Here, we provide a detailed study of the effect of a uniform
magnetization in the normal region: We show how the interplay between the
spin-momentum locking of the topological insulator and an in-plane
magnetization parallel to the direction of phase bias leads to an asymmetry of
the Andreev spectrum with respect to transverse momenta. If sufficiently large,
this asymmetry induces a transition from a regime of gapless,
counterpropagating Majorana modes to a regime with unprotected modes that are
unidirectional at small transverse momenta. Intriguingly, the
magnetization-induced asymmetry of the Andreev spectrum also gives rise to a
Josephson Hall effect, that is, the appearance of a transverse Josephson
current. The amplitude and current phase relation of the Josephson Hall current
are studied in detail. In particular, we show how magnetic control and gating
of the normal region can enable sizable Josephson Hall currents compared to the
longitudinal Josephson current. Finally, we also propose in-plane magnetic
fields as an alternative to the magnetization in the normal region and discuss
how the planar Josephson Hall effect could be observed in experiments.
