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Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting.