  A prominent signature of Majorana bound states is the exotic Josephson
effects they produce, the classic example being a fractional Josephson current
with 4\pi periodicity in the phase difference across the junction. Recent work
established that topological insulator edges support a novel `magneto-Josephson
effect', whereby a dissipationless current exhibits 4\pi-periodic dependence
also on the relative orientation of the Zeeman fields in the two banks of the
junction. Here, we explore the magneto-Josephson effect in junctions based on
spin-orbit coupled quantum wires. In contrast to the topological insulator
case, the periodicities of the magneto-Josephson effect no longer follow from
an exact superconductor-magnetism duality of the Hamiltonian. We employ
numerical calculations as well as analytical arguments to identify the domain
configurations that display exotic Josephson physics for quantum-wire
junctions, and elucidate the characteristic differences with the corresponding
setups for topological insulators edges. To provide guidance to experiments, we
also estimate the magnitude of the magneto-Josephson effects in realistic
parameter regimes, and compare the Majorana-related contribution to the
coexisting 2\pi-periodic effects emerging from non-Majorana states.
