Using atomistic tight-binding theory with a configuration interaction description of Coulomb and exchange effects, we describe excitons in symmetric quantum dots in a vertical electric field to explain how field-induced manipulation of exciton orientation and phase can eliminate the anisotropic exchange, leading to a drastic reduction of the fine-structure splitting and a 90 degree rotation of polarization, similar to experiment. This reorientation and rephasing of the exciton is done plane-by-plane inside the dot without significant squeezing of the exciton.