  We study a model of a $p$-$n$ junction in single-layer graphene in the
presence of a perpendicular magnetic field and spin-orbit interactions. By
solving the relevant quantum-mechanical problem for a potential step, we
determine the exact spectrum of spin-resolved dispersive Landau levels. Close
to zero energy, we find a pair of linearly dispersing zero modes, which possess
a wave-vector-dependent spin polarization and can be regarded as quantum
analogous of spinful snake states. We show that the Rashba spin-orbit
interaction, in particular, produces a wave vector shift between the
dispersions of these modes with observable interference effects. These effects
can in principle provide a way to detect the presence of Rashba spin-orbit
interaction and measure its strength. Our results suggest that a graphene
$p$-$n$ junction in the presence of strong spin-orbit interaction could be used
as a building block in a spin field-effect transistor.
