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We prove a bijective unitary correspondence between 1) the isospectral torusof almost-periodic, absolutely continuous CMV matrices having fixed finite-gapspectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula.This latter class arises as spectrally-dependent operator M\"obius transformsof certain generating CMV matrices which are periodic up to a rotational phase;for this reason we call them "MCMV". Such matrices are related to a choice oforthogonal rational functions on the unit circle, and their correspondence tothe isospectral torus follows from a functional model in analog to that of GMPmatrices. As a corollary of our construction we resolve a conjecture of Simon;namely, that Caratheodory functions associated to such CMV matrices arise asquadratic irrationalities.

Finite-gap CMV matrices: Periodic coordinates and a Magic Formula