Abstract: We will study the action of translation operators on wavelet subspaces in higher dimensions. This action defines an equivalence relation on the set of single wavelets of $L^2(\mathbb R^n)$ associated with an arbitrary dilation matrix. The corresponding equivalence classes will be characterized in terms of the support of the Fourier transform of the wavelets. Further, we will construct explicit examples of wavelets in each of these classes. This construction shows the existence of wavelets for which the associated wavelet subspaces are invariant under various groups of translation operators.