  We introduce an equivalence relation on the set of lattices in
$\mathbb{R}^{2d}$ such that equivalent lattices share identical structures of
Gabor frames, 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 (under a mild additional
assumption) 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.
  We also investigate the equivalence classes that have separable lattices as
representatives and find that the parameter space in this case is
$d^2$-dimensional. We provide an explicit example showing that non-separable
and irrational lattices can behave exactly like separable and rational ones.
  Finally, this approach allows us to prove a higher-dimensional variant of the
Lyubarskii-Seip-Wallst\'en 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.
