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In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady-states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of this theory in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of 4 identical microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming perfect symmetry, theory predicts 4 categories of stable spatiotemporal phase-locked periodic states and 4 categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. However, we observe only 3 of the 4 phase-locked states that are predicted for the idealized homogeneous system. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity. This work demonstrates that a surprising degree of the network's dynamics are constrained by symmetry in spite of the breakdown of the assumption of homogeneity and raises the question of why heterogeneity destabilizes some symmetry predicted states, but not others., Comment: *Ian and Mike contributed equally to this work. This work is under review by Physical Review X