  Vortices are point-like topological defects in superconductors whose motion
dictates superconducting properties and controls device performance. In
superconductor-ferromagnet heterostructures, vortices interact with topological
defects in the ferromagnet such as line-like domain walls. While in previous
heterostructure generations, vortex-domain wall interactions were mediated by
stray fields; in new heterostructure families, more important become exchange
fields and spin-orbit coupling. However, spin-orbit coupling's role in
vortex-domain wall interactions remains unexplored. Here we uncover, via
numerical simulations and Ginzburg-Landau theory, that Rashba spin-orbit
coupling induces magnetoelectric interactions between vortices and domain walls
that crucially depend on the wall's winding direction$-$its helicity. The
wall's helicity controls whether vortices are pushed or dragged by N\'eel
walls, and their gliding direction along Bloch walls. Our work capitalizes on
interactions between topological defects from different order parameters and of
different dimensionality to engineer enhanced functionality.
