When an odd particle or hole is added to the core, and the time-odd fields are taken into account, it exerts polarization effects both in the time-even (mass and shape) and time-odd (spin) channels. It means that a non-zero average spin value of the odd particle induces a time-odd component of the mean field, which influences average spin values of all particles, leading to a self-consistent amplification of the spin polarization.
It should be noted at this point that the spin polarization effect dramatically depends on the assumed symmetries and choices made for the occupied orbital, see discussion in the Appendix of Ref. [48]. Indeed, without the time-odd fields, in order to occupy the odd particle or hole, one can use any linear combination of states forming the Kramers-degenerate pair. The total energy is independent of this choice, because the time-even density matrix does not depend on it. This allows for making specific additional assumptions about the conserved symmetries, e.g., in the standard case, one assumes that the odd state is an eigenstate of the signature (or simplex) symmetry with respect to the axis perpendicular to the symmetry axis. However, in order to fully allow for the spin polarization effects through time-odd fields, one has to release all such restrictive symmetries and allow for alignment of the spin of the particle along the symmetry axis. This requires calculations with broken signature symmetry and only the parity symmetry being conserved. Therefore, calculations of this kind are more difficult than those performed within the standard cranking model.