  We establish a new covariant phenomenological model, which describes an
influence of pseudoscalar (axion) field on spins of test massive particles. The
model includes general relativistic equations of particle motion and spin
evolution in background pseudoscalar (axion), electromagnetic and gravitational
fields. It describes both the direct spin-axion coupling of the gradient type
and indirect spin-axion interaction mediated by electromagnetic fields. Special
attention is paid to the direct spin-axion coupling caused by the gradient of
the pseudoscalar (axion) field. We show that it describes a spin precession,
when the pseudoscalar (axion) field is inhomogeneous and/or non-stationary.
Applications of the model, which correspond to the three types of four-vectors
attributed to the gradient of the pseudoscalar (axion) field (time-like,
space-like, and null), are considered in detail. These are the spin precessions
induced by relic cosmological axions, axions distributed around spherically
symmetric static objects, and axions in a gravitational wave field,
respectively. We discuss features of the obtained exact solutions and some
general properties of the axionically induced spin rotation.
