The mechanical response of rubbers has been ubiquitously assumed to be only a function of the imposed strain. Using innovative X-ray measurements capturing the three-dimensional spatial volumetric strain fields, we demonstrate that rubbers and indeed many common engineering polymers undergo significant local volume changes. But remarkably, the overall specimen volume remains constant regardless of the imposed loading. This strange behavior which also leads to apparent negative local bulk moduli is due to the presence of a mobile phase within these materials. Combining X-ray tomographic observations with high-speed radiography to track the motion of the mobile phase, we have revised classical thermodynamic frameworks of rubber elasticity. The work opens broad avenues to understand not only the mechanical behavior of rubbers but a large class of widely used engineering polymers.