  General permutations acting on the Haar system are investigated. We give a
necessary and sufficient condition for permutations to induce an isomorphism on
dyadic BMO. Extensions of this characterization to Lipschitz spaces $\lip,
(0<p\leq1)$ are obtained. When specialized to permutations which act on one
level of the Haar system only, our approach leads to a short straightforward
proof of a result due to E.M.Semyonov and B.Stoeckert.
