The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*-algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of s-resolving, finite-to-one factor maps, then the construction of the stable C*-algebra is contravariant, while that of the unstable C*-algebra is covariant. The paper also discusses the constructions of these C*-algebras for Smale spaces which are not mixing.