We theoretically investigate the deformation of atomic p± orbitals driven by strong elliptically polarized (EP) laser fields and the role it plays in tunnel ionization. Our study reveals that different Stark effects induced by orthogonal components of the EP field give rise to subcycle rearrangement of the bound electron density, rendering the initial p₊ and p₋ orbitals deformed and polarized along distinctively tilted angles with respect to the polarization ellipse of the EP field. As a consequence, the instantaneous tunneling rates change such that for few-cycle EP laser pulses the bound electron initially counterrotating (corotating) with the electric field is most likely released before (after) the peak of the electric field. We demonstrate that with a sequential-pulse setup one can exploit this effect to spatially separate the photoelectrons detached from p₊ and p₋ orbitals, paving the way towards robust control of spin-resolved photoemission in laser-matter interactions.