  We investigate, using first-principles methods and effective-model
simulations, the spin-orbit coupling proximity effects in a bilayer
heterostructure comprising phosphorene and WSe$_2$ monolayers. We specifically
analyze holes in phosphorene around the $\Gamma$ point, at which we find a
significant increase of the spin-orbit coupling that can be attributed to the
strong hybridization of phosphorene with the WSe$_2$ bands. We also propose an
effective spin-orbit model based on the ${\bf C}_{1{\rm v}}$ symmetry of the
studied heterostructure. The corresponding spin-orbit field can be divided into
two parts: the in-plane field, present due to the broken nonsymmorphic
horizontal glide mirror plane symmetry, and the dominant out-of-plane field
triggered by breaking the out-of-plane rotational symmetry of the phosphorene
monolayer. Furthermore, we also demonstrate that a heterostructure with
60$^\circ$ twist angle exhibits an opposite out-of-plane spin-orbit field,
indicating that the coupling can effectively be tuned by twisting. The studied
phosphorene/WSe$_2$ bilayer is a prototypical low common-symmetry
heterostructure in which the proximity effect can be used to engineer the spin
texture of the desired material.
