Spin models are considered on a discretized inner symmetry space with tetrahedral symmetry as possible dynamical schemes for the tetron model. Parity violation, which corresponds to a change of sign for odd permutations, is shown to dictate the form of the Hamiltonian. It is further argued that such spin models can be obtained from more fundamental principles by considering a (6+1)- or (7+1)-dimensional spacetime with octonion multiplication.