On global linearization of planar involutions

Let $\phi:\R^2\to\R^2$ be an orientation--preserving $C^1$ involution such that $\phi(0)=0$ and let ${\rm Spc}\,(\phi)=\{{\rm Eigenvalues\,\,of}\,\, D\phi(p)\mid p\in\R^2\}$. We prove that if ${\rm Spc}\,{(\phi)}\subset\R$ or ${\rm Spc}\,(\phi)\cap [1,1+\epsilon)=\emptyset$ for some $\epsilon>0$ then $\phi$ is globally $C^1$ conjugate to the linear involution $D\phi(0)$ via the conjugacy $h=(I+D\phi(0)\phi)/2$, where $I:\R^2\to\R^2$ is the identity map. Similarly, if $\phi$ is an orientation-reversing $C^1$ involution such that $\phi(0)=0$ and ${\rm Trace}\,\big(D\phi(0)D\phi(p)\big)>-1$ for all $p\in\R^2$ then $\phi$ is globally $C^1$ conjugate to the linear involution $D\phi(0)$ via the conjugacy $h$. Finally, we show that $h$ may fail to be a global linearization of $\phi$ if the above conditions are not fulfilled.