The function $f(x)$ is defined by \[f(x) = \left\{ \begin{array}{cl} -x + 1 & \text{if $x \le 0$}, \\ -(x - 1)^2 & \text{if $x \ge 1$}. \end{array} \right.\]Solve $f(x) = f^{-1}(x).$ Enter all the solutions, separated by commas.
The solutions are 0, 2, and 4.
