Degeneracy of the bright single exciton spin state is a prerequisite for the production of triggered polarization-entangled photon pairs from the biexciton decay of a quantum dot. Normally, however, the exciton spin states are split due to in-plane asymmetries. Here the authors demonstrate that the exciton splitting of an individual dot can be tuned through zero by thermal annealing. Repeated annealing blueshifts the exciton emission line of the dot, accompanied by a reduction and inversion in polarization splitting. Annealing is also demonstrated to control the detuning between the exciton and biexciton transitions in any selected dot.