A new group of reduced-order models (ROMs) for nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. The Eddington (quasidiffusion) tensor that provides exact closure for the system of moment equations is approximated via one of several data-based methods of model-order reduction. These methods are the (i) proper orthogonal decomposition, (ii) dynamic mode decomposition (DMD), (iii) an equilibrium-subtracted DMD variant. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be accurate even with very low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.