In recent years lines along which structure and dynamics are invariant to a good approximation, so-called isomorphs, have been identified in the thermodynamic phase diagrams of several model liquids and solids. This paper reports computer simulations of the transverse and longitudinal collective dynamics at different length scales along an isomorph of the Lennard-Jones system. Our findings are compared to corresponding results along an isotherm and an isochore. Confirming the theoretical prediction, the reduced-unit dynamics of the transverse momentum density is invariant to a good approximation along the isomorph at all time and length scales. Likewise, the wave-vector dependent shear-stress autocorrelation function is found to be isomorph invariant. A similar invariance is not seen along the isotherm or the isochore. Using a spatially non-local hydrodynamic model for the transverse momentum-density time-autocorrelation function, the macroscopic shear viscosity and its wave dependence are determined, demonstrating that the shear viscosity is isomorph invariant on all length scales studied. This analysis implies the existence of a novel length scale which characterizes each isomorph. The transverse sound-wave velocity, the Maxwell relaxation time, and the rigidity shear modulus are also isomorph invariant. In contrast, the reduced-unit dynamics of the mass density is not invariant at length scales longer than the inter-particle distance. By fitting to a generalized hydrodynamic model, we extract values for the wave-vector-dependent thermal diffusion coefficient, sound attenuation coefficient, and adiabatic sound velocity. The isomorph variation of these quantities in reduced units at long length scales can be eliminated by scaling with $gamma$, a fundamental quantity in the isomorph theory framework, an empirical observation that remains to be explained theoretically.
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