In Riemannian or semi-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 general relativity, geodesics are the trajectories of test particles, and they can intersect in ways that imply curvature.
In the phase space for a Hamiltonian system, the integral curves never intersect, which is physically because preparing a physical system in a definite state is supposed to determine its future time evolution according to the laws of physics. Therefore(As in general relativity, the curves can be the trajectories of particles through phase space, but a point in phase space doesn't just tell you the particle's position, it tells you its entire physical state.) Since the integral curves can't intersect, 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.