Extreme radiative phenomena, where the radiation energy density and flux strongly influence the medium, are common in the universe. Nevertheless, because of limited or nonexistent observational and experimental data, the validity of theoretical and numerical models for some of these radiation-dominated regimes remains to be assessed. Here, we present the theoretical framework of a new class of laboratory astrophysics experiments that can take advantage of existing high-power laser facilities to study supersonic radiation-dominated waves. Based on an extension of Lie symmetry theory we show that the stringent constraints imposed on the experiments by current scaling theories can in fact be relaxed, and that astrophysical phenomena can be studied in the laboratory even if the ratio of radiation energy density to thermal energy and systems’ microphysics are different. The validity of this approach holds until the hydrodynamic response of the studied system starts to play a role. These equivalence symmetries concepts are demonstrated using a combination of simulations for conditions relevant to Type I X-ray burst and of equivalent laboratory experiments. These results constitute the starting point of a new general approach expanding the catalog of astrophysical systems that can be studied in the laboratory.