A separable and metrizable space X is a matchbox manifold if each point x of X has an open neighborhood which is homeomorphic to 4 x 1 for some zero-dimensional space Sx . Each arc component of a matchbox manifold admits a parameterization by the reals 1 in a natural way. This is the main tool in defining the orientability of matchbox manifolds. The orientable matchbox manifolds are precisely the phase spaces of one-dimensional flows without rest points. We show in this paper that a compact homogeneous matchbox manifold is orientable. As an application a new proof is given of Hagopian's theorem that a homogeneous metrizable continuum whose only proper nondegenerate subcontinua are arcs must be a solenoid. This is achieved by combining our work on matchbox manifolds with Whitney's theory of regular curves.