We introduce an equivalence relation on the set of lattices in \({\mathbb {R}}^{2d}\) such that equivalent lattices support identical structures of Gabor systems, up to unitary equivalence—a notion we define. These equivalence classes are parameterized by symplectic forms on \({\mathbb {R}}^{2d}\) and they consist of lattices related by symplectic transformations. This implies that \(2d^2 - d\) parameters suffice to describe the possible structures of Gabor systems over lattices in \({\mathbb {R}}^{2d}\) , as opposed to the \(4d^2\) degrees of freedom in the choice of lattice. We also prove that (modulo a minor complication related to complex conjugation) symplectic transformations are the only linear transformations of the time-frequency plane which implement equivalences of this kind, thereby characterizing symplectic transformations as the structure-preserving transformations of the time-frequency plane in the context of Gabor analysis. In addition, we investigate the equivalence classes that have separable lattices \(A_1 {\mathbb {Z}}^d\times A_2 {\mathbb {Z}}^d\) as representatives and find that the parameter space in this case is \(d^2\) -dimensional. We provide an explicit example showing that non-separable lattices with irrational lattice points can behave exactly like separable and rational ones. This approach also allows us to formulate and prove a higher-dimensional variant of the Lyubarskii-Seip-Wallstén Theorem for Gaussian Gabor frames. This gives us, for a large class of lattices in \({\mathbb {R}}^{2d}\) (including all symplectic ones), necessary and sufficient conditions for d-parameter families of Gaussians to generate Gabor frames.
