Belousov-Zhabotinsky (BZ) autonomic hydrogel composites contain active nodes of immobilized catalyst (Ru) encased within a nonactive matrix. Designing functional hierarchies of chemical and mechanical communication between these nodes enables applications ranging from encryption, sensors, and mechanochemical actuators to artificial skin. However, robust design rules and verification of computational models are challenged by insufficient understanding of the relative importance of local (molecular) heterogeneities, active node shape, and embedment geometry on transient and steady-state behavior. We demonstrate the predominance of asymmetric embedment and node shape in low-strain, BZ-gelatin composites and correlate behavior with gradients in BZ reactants. Asymmetric embedment of square and rectangular nodes results in directional steady-state waves that initiate at the embedded edge and propagate toward the free edge. In contrast, symmetric embedment does not produce preferential wave propagation because of a lack of diffusion gradient across the catalyzed region. The initiation at the embedded edge is correlated with bromide absorption by the inactive matrix, which locally elevates the bromate concentration required for catalyst oxidation. The competition between embedment asymmetry and node geometry was used to demonstrate a repeatable switch in wave direction that functions as a signal delay. Furthermore, signal propagation in or out of the composite was demonstrated via embedment asymmetry and relative dimensions of a T-shaped active network node. Overall, structural asymmetry provides a robust approach to controlling initiation and orientation of chemical-mechanical communication within composite BZ gels.