  The nature and symmetry of the transition mechanisms in the spin-spiral
copper halides CuCl$_2$ and CuBr$_2$ are analyzed theoretically. The
magnetoelectric effects observed in the two multiferroic compounds are
described and their phase diagram at zero and applied magnetic fields are
worked out. The emergence of the electric polarization at zero field below the
paramagnetic phase is shown to result from the coupling of two distinct
spin-density waves and to be only partly related to the Dzialoshinskii-Moriya
interactions. Applying a magnetic field along the two-fold monoclinic axis of
CuCl$_2$ yields a decoupling of the spin-density waves modifying the symmetry
of the phase and the spin-spiral orientation. The remarkable periodic
dependences of the magnetic susceptibility and polarization, on rotating the
field in the monoclinic plane, are described theoretically.
