This paper demonstrates that the geometry and topology of material lines in time-periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality,. This property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the Jacobian matrix of the n-period Poincare map associated with the flow. Asymptotic directionality also determines the topology of the invariant unstable manifolds of the Poincare map, which are everywhere tangent to the held of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the invariant unstable manifolds associated with a Poincare section to any desired level of detail. Since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the flow (this concept is introduced in this article), such reconstructions can be used to characterize the topological and statistical properties of partially mixed structures quantitatively. Asymptotic directionality provides evidence of a global self-organizing structure characterizing physically realizable chaotic mixing systems which is analogous to that of Anosov diffeomorphisms, which turns out to represent the basic prototype of a mixing system. In this framework we show how partially mixed structures can be quantitatively characterized by a non-uniform stationary measure (different from the ergodic measure) associated with the dynamical system generated by the field of asymptotic unstable eigenvectors.