  The M\"ossbauer rotor effect recently gained a renewed interest due to the
discovery and explanation of an additional effect of clock synchronization
which has been missed for about 50 years, i.e. starting from a famous book of
Pauli, till some recent experimental analyses. The theoretical explanation of
such an additional effect is due to some recent papers in both the general
relativistic and the special relativistic frameworks. In the first case
(general relativistic framework) the key point of the approach is the
Einstein's equivalence principle (EEP), which, in the words of the same
Einstein, enables "the point of view to interpret the rotating system K' as at
rest, and the centrifugal field as a gravitational field". In this paper, we
analyse both the history of the M\"ossbauer rotor effect and its interpretation
from the point of view of Einstein's general theory of relativity (GTR) by
adding some new insight. In particular, it will be shown that, if on one hand
the "traditional" effect of redshift has a strong analogy with the
gravitational redshift, on the other hand the additional effect of clock
synchronization has an intriguing analogy with the cosmological redshift.
Finally, we show that a recent claim in the literature that the second effect
of clock synchronization does not exist is not correct.
