Nonlinear differential matrix equations generally stem from the semi-discretization on a rectangular grid of nonlinear partial differential equations (PDEs). The two main challenges related to approximating the solution of such matrix equations includes the high computational cost of time integrating the system when the matrices have large dimensions, as well as the cost related to evaluating the time-dependent nonlinear term at each timestep. In the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to an effective, structure aware low order approximation of the original problem. The nonlinear term is also reduced by means of a fully matricial interpolation using left and right projections onto two distinct reduction spaces, giving rise to a new two-sided version of DEIM. Several numerical experiments based on typical benchmark problems illustrate the effectiveness of the new matrix-oriented setting.