We study in detail self-assembled (In,Ga)As quantum dots grown on GaP substrate from the structural, theoretical, and optical points of view. Single quantum dot morphology is first determined at the atomic level using plane-view scanning tunneling microscopy. Unusual C2 symmetry properties with high-index {136} facets are demonstrated for small quantum dots, whereas the apparent shape of the quantum dots approximately exhibits C2v symmetry, with the appearance of low-index {111} facets when the quantum dot ripens. This is interpreted as a consequence of the competition between strain and surface energy during quantum dot formation. Electronic properties are then simulated using both k·p and tight-binding models. The indium content and geometry of the quantum dots are found to have a strong influence on the transition type (direct-indirect). Finally, temperature-dependent optical properties of quantum dots are analyzed between 10 and 375 K. Photoluminescence and time-resolved photoluminescence studies show a clear proximity of two different types of optical transitions. Supported by the theoretical calculations, these transitions are interpreted as a competition between conduction band states in the X and Γ valleys.