Spin and pseudospin in graphene are known to interact under enhanced spin-orbit interaction giving rise to an in-plane Rashba spin texture. Here we show that Au-intercalated graphene on Fe(110) displays a large (∼230 meV) bandgap with out-of-plane hedgehog-type spin reorientation around the gapped Dirac point. We identify two causes responsible. First, a giant Rashba effect (∼70 meV splitting) away from the Dirac point and, second, the breaking of the six-fold graphene symmetry at the interface. This is demonstrated by a strong one-dimensional anisotropy of the graphene dispersion imposed by the two-fold-symmetric (110) substrate. Surprisingly, the graphene Fermi level is systematically tuned by the Au concentration and can be moved into the bandgap. We conclude that the out-of-plane spin texture is not only of fundamental interest but can be tuned at the Fermi level as a model for electrical gating of spin in a spintronic device.