The sorption behavior of two highly swelling "rubbery" polymers, natural rubber and polyethylene, has been studied. The polymers are in many aspects very different. Yet, when the solute mass uptake, in this case limonene, is plotted as a function of the square root of time, both polymers show similar "sigmoidal"-types of curves. This triggered the research to determine what mechanisms were responsible for the observed similarities and if the degree in sigmoidal behavior and swelling anisotropy could be easily assessed explicitly by any mechanical parameter. It was found that their degrees of swelling anisotropy, described by a solute-surface-concentration relaxation time, could be explained by their relative bulk moduli. It was assumed that the ratio in bulk modulus at zero pressure, determined from compression measurements, could represent the ratio in expansion bulk modulus during swelling. However, the prediction in swelling anisotropy during sorption using the ratio in bulk modulus was slightly less successful when the swelling anisotropy was quantified as the relative ratio of sheet thickness to cross-sectional area side length. It should be noted that the ratio in uniaxial tensile modulus between polyethylene and natural rubber was several orders of magnitude higher than their ratio in swelling anisotropy. The natural rubber sheet became saddle-shaped during limonene sorption and collapsed into a flat shape when the saturation concentration was approached. During desorption, the sheet went from flat to cup-shaped and then flat again at the end of desorption. The saddle and cup shapes occurred in both square and round sheets. These shapes are believed to be a consequence of buckling and deformation due to instabilities in the stress state of the sheet This was, in turn, explained by the normally existing local variation in cross-link density.