The Ising model serves as a canonical platform for exploring emergent symmetry in quantum critical systems. The critical point of the 1D Ising chain is described by a conformal Ising field theory, which remains integrable in the presence of a magnetic perturbation, leading to massive particles associated with the exceptional Lie algebra $E_8$. Weakly coupling two Ising chains into a ladder breaks this integrability and is predicted to bind the elementary kink excitations of the decoupled chains, giving rise to a richer spectrum of excitations described by the ${𝒟}_8^{(1)}$ theory. Experimental signatures of $E_8$ excitations have arguably been observed in scattering studies of the spin chain material CoNb${}_2$O${}_6$, but direct observation of the bound-state spectrum emerging from inter-chain coupling has remained elusive. Here, we probe these emergent symmetries in a Rydberg atom quantum processing unit, leveraging its tunable geometry to realize both chain and ladder configurations. We identify mass spectra consistent with $E_8$ at the single-chain critical point and, in the weakly coupled ladder, report the first signatures of the reorganization of Ising excitations into the bound-state spectrum predicted by ${𝒟}_8^{(1)}$ symmetry. Our results demonstrate the power of Rydberg platforms for investigating symmetry emergence in quantum many-body systems and provide a direct window into the interplay of quasiparticle interactions, geometry, and criticality.