In this paper we consider crossed product algebras of piecewise constant function algebras on the real line that arise in multiresolution analysis. Such algebras form an increasing sequence of algebras of functions on the real line. We derive conditions under which these algebras are invariant under a bijection on the real line, in which case we get an increasing sequence of crossed product algebras. We then give a comparison of commutants (centralizers) in a number of cases.