  The dynamics of spin in external electromagnetic, gravitational, and axion
fields is analysed in the framework of the gravitoelectromagnetism approach in
Einstein's general relativity theory. We consistently extend the recent studies
from the flat Minkowski geometry to the curved spacetime manifolds,
contributing to the discussion of the possible new role of a precessing spin as
an ``axion antenna'' that can be used to detect the hypothetical axion-like
dark matter. The formalism developed helps to clarify the subtle influence of
the gravitational/inertial and axion fields in the ultra-sensitive high-energy
spin experiments with charged particles and neutrons at accelerators and
storage rings devoted to testing fundamental physical symmetries, including
attempts to establish the nature of dark matter in the Universe.
