Reducing conjugacy in the full diffeomorphism group of R to conjugacy in the subgroup of orientation-preserving maps

Let $\text{Diffeo}=\text{Diffeo}(\mathbb{R})$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\mathbb{R}$, under the operation of composition, and let ${\text{Diffeo}}^{+}$ be the subgroup of diffeomorphisms of degree +1, i.e. orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements $f,g\in \text{Diffeo}$ are conjugate in $\text{Diffeo}$ to associated conjugacy problems in the subgroup ${\text{Diffeo}}^{+}$. The main result concerns the case when $f$ and $g$ have degree -1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in ${\text{Diffeo}}^{+}$, in order to ensure that $f$ is conjugated to $g$ by an element of ${\text{Diffeo}}^{+}$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval.