Non-equilibrium dynamics is omnipresent in nature and technology and can exhibit symmetries and order. In idealistic systems this universality is well-captured by traditional models of dynamical systems. Realistic processes are often more complex. This work considers two paradigmatic complexities—canonical Kolmogorov turbulence and interfacial Rayleigh-Taylor mixing. We employ symmetries and invariant forms to assess very different properties and characteristics of these processes. We inter-link, for the first time, to our knowledge, the scaling laws and spectral shapes of Kolmogorov turbulence and Rayleigh-Taylor mixing. We reveal the decisive role of the control dimensional parameters in their respective dynamics. We find that the invariant forms and the control parameters provide the key insights into the attributes of the non-equilibrium dynamics, thus expanding the range of applicability of dynamical systems well-beyond traditional frameworks.