The transfer matrices of solvable lattice spin models are considered under general boundary conditions associated with the symmetry of the models. It is shown how the full symmetry can be exploited for systems having a sublattice symmetry by considering an extended model where the states distinguish the sublattices. In this extended scheme, the parity of the number of sites is tied to the boundary conditions. The inversion identities can be extended and be used to obtain the full operator content of the critical models. As an application the complete spectra of the transfer matrices for the eight-vertex model at the decoupling point are calculated.