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