  Starting from the irreducible representations of the group of the wave vector
we construct the spin wave functions consistent with inversion symmetry,
neglected in the usual representation analysis. We obtain the relation between
the basis functions of different members of the star of the wave vector. We
introduce order parameters and determine their transformation properties under
the operations of the space group of the paramagnetic crystal. The results are
applied to construct terms in the magnetoelectric interaction which are
quadratic and quartic in the magnetic order parameters. The higher order
magnetoelectric interactions can in principle induce components of the
spontaneous polarization which are not allowed by the lowest order
magnetoelectric interaction. We also obtain the relation between the spin wave
functions of the incommensurate phase and those of the commensurate phase which
lead to analogous relations between the order parameters of these two phases.
