Electrons in atoms possess both spin and orbital degrees of freedom. In non-relativistic quantum mechanics, these are independent, resulting in large degeneracies in atomic spectra. However, relativistic effects couple the spin and orbital motion, leading to the well-known fine structure in their spectra. It is widely believed that the electronic states of defect-free carbon nanotubes are four-fold degenerate, owing to independent spin and orbital symmetries, and also possess electron-hole symmetry. In this talk I will show our recent measurements, which demonstrate that in ultra-clean nanotubes the spin and orbital motion of electrons are coupled, thereby breaking all of these symmetries. This spin-orbit coupling is directly observed as a splitting of the four-fold degeneracy of a single electron in ultra-clean quantum dots. It further breaks the electron-hole symmetry by aligning the orbital and spin magnetic moments differently for electrons and holes. Our observations are consistent with recent theories, which predict that in the cylindrical topology of nanotubes, the motion of electrons along closed orbits would be coupled to their spin. These findings have important implications on our basic understanding of the electronic properties of nanotubes as well as on the future use of carbon-based systems for spin-based applications.