Driven far away from equilibrium by both uniform and random external fields, a system of diffusing particles with short-range attractive forces displays many singular thermodynamic properties. Surprisingly, measuring pair correlations in lattice-gas models with saturation drives, we find little difference between the uniform and random cases, even though the underlying symmetries are quite distinct. Motivated by this puzzle, we study three-point correlations using both field-theoretic and simulation techniques. The continuum theory predicts the following: (a) The three-point function is nonzero only for the uniformly driven system; (b) it is odd under a parity transformation; and (c) there exists an infinite discontinuity singularity at the origin in momentum space. Simulation results are clearly consistent with these predictions. Based on these findings, we suggest several avenues for future investigations.