We define a family of natural decompositions of Sturmian words in Christoffel words, called reversible Christoffel (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always) arbitrarily long Abelian equivalent prefixes. Using the three gap theorem, we prove that in each RC factorization, only 2 or 3 distinct Christoffel words may occur. We begin the study of such factorizations, considered as infinite words over 2 or 3 letters, and show that in the general case they are either Sturmian words, or obtained by a three-interval exchange transformation.