We present a detailed study of biexciton complexes formed in single, site-controlled pyramidal GaInAs/GaAs quantum dots (QDs). By using power dependent measurements and photon correlation spectroscopy, we identify the excitonic transitions of a large number of pyramidal QDs, exhibiting both positive and negative biexciton binding energies. Separation of charges within the QD, caused by piezoelectric fields, is believed to be responsible for the positive to negative crossover of the biexciton binding energy with increasing QD size. In particular, QDs exhibiting vanishing biexciton binding energies are evidenced, with potential applications in quantum information processing. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4765646]