The dynamics of the neutron spin in whispering gallery states is considered in the case of the presence of hypothetical spin-dependent interactions between nucleons (axion-like interactions). It is shown that such interaction generates an additional field affecting the spatial and spin variables of the neutron. The reveal of this field in the scattering and polarisation of neutrons on a cylindrical mirror in whispering gallery states, the sensitivity to fifth forces of physical observables in different types of scattering experiments are discussed.