The paper investigates an extension of Christoffel duality to a certain family of Sturmian words. Given an Christoffel prefix w of length N of an Sturmian word of slope g we associate a N-companion slope gN∗ such that the upper Sturmian word of slope gN∗ has a prefix w∗ of length N which is the upper Christoffel dual of w. Although this condition is satisfied by infinitely many slopes, we show that the companion slope gN∗ is an interesting and somewhat natural choice and we provide geometrical and music-theoretical motivations for its definition.In general, the second-order companion (gN∗)N∗=gN∗∗ does not coincide with the original g. We show that, given a rational number 0<MN<1, the map g→gN∗∗ has exactly one fixed point, ϕMN∈[0,1), called odd mirror number. We show that odd mirror numbers are Sturm numbers and their continued fraction expansion is purely periodic with palindromic periods of even length. The semi-periods are of odd length and form a binary tree in bijection to the Farey tree of ratios 0<MN<1. Its root is the singleton {2}, which represents the odd mirror number −1+82=[0;22¯]. The characteristic word cϕMN of slope ϕMN remains fixed under a standard morphism which can be computed from the semi-period of ϕMN. Finally, we prove that the characteristic word G(cϕMN) is a harmonic word.As a minor open question we ask for the properties of even mirror numbers. A final conjecture provides a proper word-theoretic meaning to the extended duality for odd mirror number slopes: given a characteristic word cϕMN, the succession of those letters which immediately precede the occurrences of the left special factor of length N coincides–up to letter exchange–with the G-image of the dual word cϕMN∗.