Understanding and classifying non-equilibrium many-body phenomena, analogously to the classification of equilibrium states of matter into universality classes is an outstanding problem in physics. From stellar matter to financial markets, any many-body system can be out of equilibrium in a myriad of ways, and many are difficult to experiment on. It is therefore a major goal to establish universal principles that apply to different phenomena and physical systems. For equilibrium states, the universality seen in the self-similar spatial scaling of systems close to phase transitions lies at the heart of their classification. Recent theoretical work and experimental evidence suggest that isolated many-body systems far from equilibrium generically exhibit dynamic (spatiotemporal) self-similar scaling, akin to turbulent cascades and the Family–Vicsek scaling in classical surface growth. Here we observe bidirectional dynamic scaling in an isolated quench-cooled atomic Bose gas; as the gas thermalizes and undergoes Bose–Einstein condensation, it shows self-similar net flows of particles towards the infrared (smaller momenta) and energy towards the ultraviolet (smaller length scales). For both infrared and ultraviolet dynamics we find that the scaling exponents are independent of the strength of the interparticle interactions that drive the thermalization.