  We prove a bijective unitary correspondence between 1) the isospectral torus
of almost-periodic, absolutely continuous CMV matrices having fixed finite-gap
spectrum and 2) special periodic block-CMV matrices satisfying a Magic Formula.
This latter class arises as spectrally-dependent operator M\"obius transforms
of 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 of
orthogonal rational functions on the unit circle, and their correspondence to
the isospectral torus follows from a functional model in analog to that of GMP
matrices. As a corollary of our construction we resolve a conjecture of Simon;
namely, that Caratheodory functions associated to such CMV matrices arise as
quadratic irrationalities.
