Suppose I have a simple two dimensional integrable twist map, such as $x_{1}=x_{0}+y_{0}, \quad y_{1}=y_{0}$. Suppose that I perturb it in such a way that Moser's Twist Theorem is satisfied.
What about if I now, after perturbation, do a reflection, $x_{1} \rightarrow -x_{1}, y_{1} \rightarrow -y_{1}$.
Does Moser's Twist Theorem immediately apply here as well? Would it have to be modified in some way?
Thank you very much for the answer(s)!