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We study the involution of the real line induced by the outer automorphism ofthe extended modular group PGL(2,Z). This `modular' involution is discontinuousat rationals but satisfies a surprising collection of functional equations. Itpreserves the set of real quadratic irrationals mapping them in a non-obviousway to each other. It commutes with the Galois action on real quadraticirrationals. More generally, it preserves set-wise the orbits of the modulargroup, thereby inducing an involution of the moduli space of real rank-twolattices. We give a description of this involution as the boundary action of acertain automorphism of the infinite trivalent tree. It is conjectured thatalgebraic numbers of degree at least three are mapped to transcendental numbersunder this involution.

On the involution of the real line induced by Dyer's outer automorphism of PGL(2,Z)