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Let $\Diffeo=\Diffeo(\R)$ denote the group of infinitely-differentiablediffeomorphisms of the real line $\R$, under the operation of composition, andlet $\Diffeo^+$ be the subgroup of diffeomorphisms of degree +1, i.e.orientation-preserving diffeomorphisms. We show how to reduce the problem ofdetermining whether or not two given elements $f,g\in \Diffeo$ are conjugate in$\Diffeo$ to associated conjugacy problems in the subgroup $\Diffeo^+$. Themain 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 theassumption that their (compositional) squares are conjugate in $\Diffeo^+$, inorder to ensure that $f$ is conjugated to $g$ by an element of $\Diffeo^+$. Themethods involve formal power series, and results of Kopell on centralisers inthe diffeomorphism group of a half-open interval.

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