A thermodynamically equilibrated fluid of hard spheroids is a simple model of liquid matter. In this model, the coupling between the rotational degrees of freedom of the constituent particles and their translations may be switched off by a continuous deformation of a spheroid of aspect ratio t into a sphere (t=1). We demonstrate, by experiments, theory, and computer simulations, that dramatic nonanalytic changes in structure and thermodynamics of the fluids take place, as the coupling between rotations and translations is made to vanish. This nonanalyticity, reminiscent of a second-order liquid-liquid phase transition, is not a trivial consequence of the shape of an individual particle. Rather, free volume considerations relate the observed transition to a similar nonanalyticity at t=1 in structural properties of jammed granular ellipsoids. This observation suggests a deep connection to exist between the physics of jamming and the thermodynamics of simple fluids.