  Transition metal dichalcogenides exhibit giant spin-orbit coupling, and
intriguing spin-valley effects, which can be harnessed through proximity in van
der Waals (vdW) heterostructures. Remarkably, due to the prismatic crystal
field, the Zeeman-type band splitting of valence bands reach values of several
hundreds of meV. While this effect is suppressed in the commonly studied
hexagonal (H)-stacked bilayers due to the presence of inversion symmetry, the
recent discovery of sliding ferroelectricity in rhombohedral (R-)stacked MX$_2$
bilayers (M=Mo, W; X=S, Se) suggests that the Zeeman effect could be present in
these non-centrosymmetric configurations, making it even more intriguing to
investigate how the spin-resolved bands would evolve during the phase
transition. Here, we perform density functional theory calculations
complemented by symmetry analysis to unveil the evolution of ferroelectricity
during sliding and the behavior of Zeeman splitting along the transition path.
While the evolution of the out-of-plane component of the electric polarization
vector resembles the conventional ferroelectric transition, we observe
significant in-plane components parallel to the sliding direction, reaching
their maximum at the intermediate state. Moreover, we demonstrate that the
R-stacked bilayers exhibit persistent Zeeman-type band splitting throughout the
transition path, allowed by the lack of inversion symmetry. Further analysis of
different stacking configurations generated by sliding along various directions
confirms that the Zeeman effect in MX$_2$, primarily arising from the polarity
of prismatic ligand coordination of the metal atom, is remarkably robust and
completely governs the spin polarization of bands, independently of the sliding
direction. This resilience promises robust spin transport in vdW based MX$_2$
bilayers, opening new opportunities for ferroelectric spintronics.
