In this paper we study scalar multivariate subdivision schemes with general integer expanding dilation matrix. Our main result yields simple algebraic conditions on the symbols of such schemes that characterize their polynomial reproduction, i.e. their capability to generate exactly the same polynomials from which the initial data is sampled. These algebraic conditions also allow us to determine the approximation order of the associated refinable functions and to choose the "correct" parametrization, i.e. the grid points to which the newly computed values are attached at each subdivision iteration. We use this special choice of the parametrization to increase the degree of polynomial reproduction of known subdivision schemes and to construct new schemes with given degree of polynomial reproduction.