This paper analyzes the problem of global rotation in general relativity (GR) theory. Simple cosmological models with rotation and expansion are presented, which give a natural explanation of the modern values of the acceleration parameter at different red shifts without involving the concepts of “dark energy” and “dark matter”. It is shown that due to the smallness of the cosmological rotation, for its detection one should use observations that do not depend on the magnitude of the angular velocity of the Universe. Such tests include the effects of the cosmic mirror and the cosmic lens. For the first time on the basis of modern electronic catalogs the search on the celestial sphere of images of our Galaxy and other galaxies is made. Viable candidates for both effects have been found.