Spin-split antiferromagnets have significance for antiferromagnetic (AFM) spintronics due to their momentum-dependent spin polarization, which can be exploited for the control and detection of the AFM order parameter. Here, we explore the polar-layer stacking of AFM-ordered van der Waals bilayers driving the emergence of reversible electric polarization and nonrelativistic spin splitting (NRSS) of their band structure. Using the spin-space group approach and first-principles calculations, we explore $M\mathrm{P}{X}_3$ ($M=\mathrm{Mn}$ V; $X=\mathrm{S}$, Se) and 1T-${\mathrm{CrTe}}_2$ as representative two-dimensional AFM materials which exhibit different types of NRSS when stacked into a polar bilayer. We demonstrate that NRSS can have both altermagnetic and nonaltermagnetic origins and elucidate symmetry requirements for NRSS to be switchable by electric polarization. We argue that the electric polarization switching of NRSS in polar AFM bilayers may be more practical for device applications than the current-induced Néel vector switching.

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• Altermagnetism
• Antiferromagnetism
• Electric polarization
• Spin texture
• Spintronics
• 2-dimensional systems
• Altermagnets
• Antiferromagnets
• Transition metal dichalcogenides
• Transition metal trichalcogenides
• First-principles calculations
• Symmetries in condensed matter