We detail the construction of a family of lattice gas automata based on a model of 't Hooft, proceeding by use of symmetry principles to define first the kinematics of the model and then the dynamics. A spurious conserved quantity appears; we use it to effect a radical transformation of the model into one whose spacetime configurations are equivalent to the two-dimensional states of an exactly solvable statistical mechanics model, the symmetric eight-vertex model with parameters restricted to a disorder variety. We comment on the implications of this identification for the original lattice gas.