The quantum mechanical features of Hamiltonian cellular automata (CA) are reviewed, i.e. of CA described by integer valued variables and couplings that follow linear updating rules. We then discuss whether, in this class of CA, there is room for single or multi-partite systems that evolve by permutations of a set of ontological states, thus providing examples for G. 't Hooft's recent CA Interpretation of Quantum Mechanics. This is indeed the case for systems consisting of a single or multiple two-state components, interacting similarly as in the Heisenberg model of a ferromagnet. Initial conditions play a decisive role in avoiding the formation of unwanted superpositions of these prequantum states, unlike the most common situation in quantum mechanics.