Rashba spin-orbit coupling is a quintessential spin interaction appearing in virtually any electronic heterostructure. Its paradigmatic spin texture in the momentum space forms a tangential vector field. Using first-principles investigations, we demonstrate that in twisted homobilayers and hetero-multilayers, the Rashba coupling can be predominantly radial, parallel to the momentum. Specifically, we study four experimentally relevant structures: twisted bilayer graphene (Gr), twisted bilayer WSe2, and twisted multilayers WSe2/Gr/WSe2 and WSe2/Gr/Gr/WSe2. We show, that the Rashba spin-orbit field texture in such structures can be controlled by an electric field, allowing to tune it from radial to tangential. Such spin-orbit engineering should be useful for designing novel spin-charge conversion and spin-orbit torque schemes, as well as for controlling correlated phases and superconductivity in van der Waals materials.