  We prove that any smooth foliation that admits a Riemannian foliation
structure has a well-defined basic signature, and this geometrically defined
invariant is actually a foliated homotopy invariant. We also show that foliated
homotopic maps between Riemannian foliations induce isomorphic maps on basic
Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation
is invariant under foliated homotopy equivalence.
