Recent experiments and simulations have shown that two-dimensional systems can form tetratic phases with fourfold rotational symmetry, even if they are composed of particles with only twofold symmetry. To understand this effect, we propose a model for the statistical mechanics of particles with almost fourfold symmetry, which is weakly broken down to twofold. We introduce a coefficient kappa to characterize the symmetry breaking, and find that the tetratic phase can still exist even up to a substantial value of kappa. Through a Landau expansion of the free energy, we calculate the mean-field phase diagram, which is similar to the result of a previous hard-particle excluded-volume model. To verify our mean-field calculation, we develop a Monte Carlo simulation of spins on a triangular lattice. The results of the simulation agree very well with the Landau theory.