Small-angle X-ray scattering (SAXS) and bulk volumetric measurements were performed on the perfluorinated ionomer membrane "Nafion". The swelling experiments were conducted using controlled environments of different relative humidities. The membranes were oriented using uniaxial draw, both before and after conversion of the precursor material, as well as both sequential and simultaneous biaxial draw. Using a novel model-independent maximum entropy method, it is shown that the most statistically probable scattering model for Nafion is an ion clustered morphology with a hierarchical scale of structure. The smallest scale is comprised of arrays of roughly spherical ionic clusters. The clusters are agglomerated into higher-order structures, whose shape is determined by the spatial coherence of the clusters, as inferred from the SAXS measurements. Anomalies in the magnitudes of the microscopic and macroscopic swellings are shown to be caused by the rearrangement of ionic material, producing changes in the number density of clusters.