  Consider a non-negative, self-adjoint, maximally subelliptic operator on a
compact manifold. We show that the spectral multiplier is a singular integral
operator under an appropriate Mihlin-H\"ormander type condition. We establish
the equivalence between non-isotropic Besov and Triebel-Lizorkin spaces adapted
to the operator and those adapted to a Carnot-Carath\'eodory geometry on the
manifold. We also give a Mihlin-H\"ormander type condition for the boundedness
of the spectral multiplier on non-isotropic $L^p$ Sobolev spaces.
