In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition(POD)/discrete empirical interpolation method that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method, and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem.