A novel reduced-order modeling methodology is developed based on a low-dimensional representation of the solution on a coarse, reduced grid. In contrast to widely used POD-based reduced-solution representations, which have a modal structure, our solution has a nodal structure. The nodal structure of our solution allows us to mimic well-established methods for the structure-preserving high-fidelity discretization of conservation laws at a reduced level. This comes with several benefits. Most important to us is that it naturally results in global and local structure-preservation properties on the reduced grid. These properties, in turn, provide the reduced order model (ROM) with stability guarantees. The nodal approach also allows for straightforward implementation of boundary conditions. The ROM is built with the use of a reduced nodal basis, which prolongates the solution on the reduced grid to a high-fidelity grid where all computations are performed. In particular, we construct a nodal reduced basis by solving the so-called Procrustes problem. This is an optimization problem that finds a basis that optimally reconstructs a target data matrix given a predefined set of reduced coefficients. By using nodal values as reduced coefficients in this optimization problem, our basis takes a nodal structure. However, the nodal basis obtained from the Procrustes problem need not be exactly interpolatory; this can be ensured by applying a near-identity basis transformation. The nodal values we use are derived from Carathéodory-Tchakaloff (CT) theory, which provides a positive, low-dimensional empirical quadrature. We use the nodal basis and empirical CT-quadrature to construct empirical diagonal-norm upwind summation-by-parts (SBP) operators on the reduced grid, while boundary terms are handled in a stable manner using simultaneous approximation terms (SATs). Using an appropriate numerical flux function, the resulting SBP-SAT reduced-order model can preserve conservative and non-conservative secondary structures, such as entropy and kinetic energy, respectively, for the compressible Navier-Stokes equations.