In this contribution, we analyze both the old Mössbauer experiments in a rotating system and our new experiment on this subject, which unambiguously indicate the presence of an additional component in
the relative energy shift DE/E between emission and absorption lines, as compared with the classic relativistic expression written to the accuracy c^2 (that is DE/E = -u^2 /2c^2, where u is the tangential velocity of absorber, and c the light velocity in vacuum). The additional dilation of time for the rotating absorber constitutes more than 20% from the relativistic value, and it many times exceeds the measuring uncertainty. We discuss a possible origin of this effect and the ways of its further experimental verification.