In the present paper, we study the controllability of the control system associated to rolling without slipping or spinning of a Riemannian manifold (M, g) onto the hyperbolic n-space H n. Our main result states that the system is completely controllable if and only if (M, g) is not isometric to a warped product of a special form, in analogy to the classical de Rham decomposition theorem for Riemannian manifolds. The proof is based on the observations that the control-lability issue in this case reduces to determine whether (M, g) admits a reducible action of a hyperbolic analog of the holonomy group and a well-known fact about connected subgroups of O(n, 1) acting irreducibly on the Lorentzian space R n,1 .