Let $\O=\R^+$, $\mu(dx)=e^{-x}dx$ and $\theta(x)=-\log|1-2e^{-x}|$. Then $\mu$ is $\theta$-invariant.
Put $F:(0,1)\rar\R^+$, $F(t)=-\log(1-t)$, then $F^{-1}(x)=1-e^{-x}$ and
$$
F^{-1}\circ\theta\circ F(x)
=\left\{\begin{array}{cl}
2x&\mbox{if $x < 1/2$}\\
2(1-x)&\mbox{if $x\geq1/2$}
\end{array}\right.
$$