We have theoretically explored the spin-orbit interaction in carbon nanotubes. We have found that, besides the dependence on chirality and diameter, the effects of spin-orbit coupling are anisotropic: spin splitting is larger for the higher valence or the lower electron band depending on the specific tube. Different tube behaviors can be grouped in three families, according to the so called chiral index. Curvature-induced changes in the orbital hybridization have a crucial role, and they are shown to be family-dependent. Our results explain recent experiments without invoking external fields.