We present photoluminescence studies of the molecular neutral biexciton-exciton spectra of individual vertically stacked InAs/GaAs quantum dot pairs. We tune either the hole or the electron levels of the two dots into tunneling resonances. The spectra are described well within a few-level, few-particle molecular model. Their properties can be modified broadly by an electric field and by structural design, which makes them highly attractive for controlling nonlinear optical properties.