The interplay of superconductivity with non-trivial spin textures is promising for the engineering of non-Abelian Majorana quasiparticles. Spin-orbit coupling is crucial for the topological protection of Majorana modes as it forbids other trivial excitations at low energy but is typically intrinsic to the material(1-7). Here, we show that coupling to a magnetic texture can induce both a strong spin-orbit coupling of 1.1 meV and a Zeeman effect in a carbon nanotube. Both of these features are revealed through oscillations of superconductivity-induced subgap states under a change in the magnetic texture. Furthermore, we find a robust zero-energy state-the hallmark of devices hosting localized Majorana modes-at zero magnetic field. Our findings are generalizable to any low-dimensional conductor, and future work could include microwave spectroscopy and braiding operations, which are at the heart of modern schemes for topological quantum computation.