  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.
