We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $\mathbb{R}^d$-valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $L^p$, for $1\leq p<\infty$, and that they are self-adjoint when $p=2$. When the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class.