A non-singular, invertable, ergodic transformation is the quadriple $(X,\mathcal B, \mu, T)$ where $(X,\mathcal B, \mu)$ is a measure space and $T$ is an invertable, measurable automorphism where $\mu$ and $\mu\circ T$ equivalent measures.

Two such systems $(X,\mathcal B, \mu, T)$ and $(Y,\mathcal C, \nu, S)$ are isomorphic when there exists isomorphism $\phi: X \mapsto Y$ where $$ S\phi x = \phi T x$$

It seems it should be obvious that $(X,\mathcal B, \mu, T)$ and $(X,\mathcal B, \mu, T^{-1})$ are isomorphic. If I assume $X$ is a product space and $T$ is the $+1$ odometer action then I can prove it myself. But I would rather quote someone and be done with it.

Do you know of a source to quote for this?