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Spin-dependent dispersion and Rashba effect are manifestations of spin-orbit interaction (SOI) associated with the breaking of the spatial inversion symmetry in condensed matter and in optical systems. Here, we report an extraordinary spin-split dispersion effect of leaky surface plasmons in an inversion-symmetric plasmonic crystal. Momentum domain polarization analysis identified a transverse momentum-dependent linear birefringence-linear dichroism (geometric LB-LD) effect responsible for this phenomena. It is demonstrated that evolution of strong geometrical phase gradient due to the geometric LB-LD effect mediates the spin-split dispersion in inversion symmetric plasmonic metasurface through spin (circular polarization)-dependent transverse momentum. This unconventional SOI phenomena offers a new paradigm of spin-based dispersion engineering and spin-controlled nano-optical functionalities by encompassing the forbidden class of symmetric metastructures in the realm of spin optics.