The ethnic names associated with the diatonic modes (e.g., Dorian, Lydian, et al.) were assigned in one way by the ancient Greeks and in a different way in the anonymous medieval treatise Alia musica. Music historians usually say that this renaming was the result of a confusion, and leave it at that, but Edward Gollin showed that there was a logic here that could be captured in transformational terms. In this paper we add/uncover another layer to the palimpsest with the observation that the respective Greek and medieval nomenclatures are correlated with one of the fundamental distinctions in mathematical scale theory/word theory, the distinction between plain and twisted adjoint folding patterns of modes, as represented by conjugacy classes of Christoffel words.