An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.
Anthony G. O’Farrell. Ian Short. "Reversibility in the Diffeomorphism Group of the Real Line." Publ. Mat. 53 (2) 401 - 415, 2009.