Please suppose that we have the 2D non-linear discrete dynamical system(or recurrence relations) $x_{n+1}=f(x_n,y_n)$ and $y_{n+1}=g(x_n,y_n)$ where $x_n, y_n \in \mathbb{R}$ . Here, we know that the equation $f(y_{n+1},x_{n+1})=y_n$ and $g(y_{n+1},x_{n+1})=x_n$ are satisfied.($i.e.$ the inverse map of this system is given by interchanging variables and functions themselves)
Is this type of system is special or well-known? If so, please give me some reference and so on if possible.