Symmetry properties of different solutions for singlet and triplet states of some simple molecules are investigated in the framework of the half-projected Hartree-Fock (HPHF) method. HPHF initial-guess functions were obtained by using mixtures with, or substitutions by, virtual orbitals of restricted Hartree-Fock (RHF) occupied orbitals. The relation existing between the solutions and the spatial symmetry of the initial guess has been established. The elaborated criteria have been tested for the lowest singlet and triplet states of CH2, CF2 and CHF.