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

Let $\operatorname{Diffeo}=\operatorname{Diffeo}(\mathbb R)$ denote the group of infinitely-differentiable diffeomorphisms of the real line $\mathbb R$, under the operation of composition, and let $\operatorname{Diffeo}^+$ 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 $f,g\in\operatorname{Diffeo}$ are conjugate in $\operatorname{Diffeo}$ to associated conjugacy problems in the subgroup $\operatorname{Diffeo}^+$. 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 $\operatorname{Diffeo}^+$, in order to ensure that $f$ is conjugated to $g$ by an element of $\operatorname{Diffeo}^+$. The methods involve formal power series, and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibl. – 4 titles.