The optical response of (InGa)(AsSb)/GaAs quantum dots (QDs) grown on GaP (001) substrates is studied by means of excitation and temperature-dependent photoluminescence (PL), and it is related to their complex electronic structure. Such QDs exhibit concurrently direct and indirect transitions, which allows the swapping of Γ and L quantum confined states in energy, depending on details of their stoichiometry. Based on realistic data on QD structure and composition, derived from high-resolution transmission electron microscopy (HRTEM) measurements, simulations by means of k · p theory are performed. The theoretical prediction of both momentum direct and indirect type-I optical transitions are confirmed by the experiments presented here. Additional investigations by a combination of Raman and photoreflectance spectroscopy show modifications of the hydrostatic strain in the QD layer, depending on the sequential addition of QDs and capping layer. A variation of the excitation density across four orders of magnitude reveals a 50 meV energy blueshift of the QD emission. Our findings suggest that the assignment of the type of transition, based solely by the observation of a blueshift with increased pumping, is insufficient. We propose therefore a more consistent approach based on the analysis of the character of the blueshift evolution with optical pumping, which employs a numerical model based on a semi-self-consistent configuration interaction method.