In Riemannian geometry, the reason that our spaces don't all look alike in a neighborhood of a point is that parallelism fails, and the extent to which parallelism fails can be described as a curvature, which is invariant. For example, if two geodesics both start from a point and later on intersect again at some other point, this is a sign of curvature. It's invariant because intersection is invariant. It doesn't matter what coordinates you pick -- an intersection is an intersection.
In the phase space for a Hamiltonian system, the integral curves never intersect. Therefore if someone draws a set of integral curves for you, you can always bend and stretch the picture so that the integral curves look like parallel lines -- in some finite neighborhood.