  In this article we study invariance properties of shift-invariant spaces in
higher dimensions. We state and prove several necessary and sufficient
conditions for a shift-invariant space to be invariant under a given closed
subgroup of $\R^d$, and prove the existence of shift-invariant spaces that are
exactly invariant for each given subgroup. As an application we relate the
extra invariance to the size of support of the Fourier transform of the
generators of the shift-invariant space.
  This work extends recent results obtained for the case of one variable to
several variables.
