Time-periodic flow of bidisperse granular material (varying in particle size or density) in quasi-two-dimensional tumblers reveals segregation patterns that capture the symmetries of Poincaré sections, stroboscopic maps of the underlying flow, derived from a continuum model. The similarity is striking because the model does not contain information about particle type. We study this phenomenon experimentally by varying the concentration of small particles in a bidisperse mixture in quasi-two-dimensional tumblers with square and pentagonal cross-sections. By coupling experiments with an analysis of periodic points, we explain the connection between the segregation patterns and the dynamics of the underlying flow. Analysis of the eigenvectors and unstable manifolds of hyperbolic points shows the connection between regions of chaotic flow and the shape of the segregation patterns. The techniques developed here can also be applied to non-polygonal tumblers such as elliptical tumblers, circular tumblers with time-periodic forcing, and even three-dimensional tumblers forced on multiple axes of rotation. [Supported by NSF and DOE.]