In photoluminescence spectra of symmetric [111] grown GaAs/AlGaAs quantum dots in longitudinal magnetic fields applied along the growth axis, we observe in addition to the expected bright states also nominally dark transitions for both charged and neutral excitons. We uncover a strongly nonmonotonic, sign-changing field dependence of the bright neutral exciton splitting resulting from the interplay between exchange and Zeeman effects. Our theory shows quantitatively that these surprising experimental results are due to magnetic-field-induced ±3/2 heavy-hole mixing, an inherent property of systems with C_3v point-group symmetry.