I'm interested in certain involutions of the real numbers, i.e functions $f$ such that $f\circ f = \text{id}_{\mathbb{R}}$. It has been shown here that $\text{id}_{\mathbb{R}}$ is the only increasing continuous involution, and that the decreasing involutions are the conjugates of $-\text{id}_{\mathbb{R}}$, i.e the $\phi^{-1}\circ -\text{id}_{\mathbb{R}} \circ \phi$ for $\phi$ a continuous bijection of the reals.
An infinite family of smooth (infinitely differentiable) involutions is given by the $a - \text{id}_{\mathbb{R}}$ for $a \in \mathbb{R}$, which I'll call trivial because they're boring.
I managed to construct the following involution that is $\mathcal{C}^2$ : $$f(x) = \begin{cases}-x^2-x &\text{if } x\ge 0 \\ \frac{\sqrt{1-4x}-1}{2} &\text{if } x<0\end{cases}$$
Do there exist any non-trivial $\mathcal{C}^\infty$ involutions? I haven't had any luck finding any so far.
Thank you in advance.