We substantiate the existence of the new type of rotation of a diatomic molecule with identical nuclei in which the nuclei periodically exchange their spatial coordinates. In the exchange rotation, the electron shell of the molecule preserves its orientation in space. The exchange rotation is considered as a particular case of molecular free rotation. We analyze the experiment on spontaneous Raman scattering of laser radiation by hydrogen molecules inside liquid water (D. G. Taylor and H. L. Strauss, J. Chem. Phys. 90(2), 1989). The intensity of scattered light and its distribution over Stokes components is shown to correspond to the assumption of the exchange rotation of hydrogen molecules in a liquid.