The photoluminescence of ${\mathrm{In}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{A}\mathrm{s}/\mathrm{A}\mathrm{l}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ quantum disks is investigated by microscope photoluminescence (PL) and PL excitation spectroscopies. The excited-state transitions of excitons and biexcitons are studied in terms of their dependence on excitation photon polarization. Their origins are analyzed on the basis of symmetry properties expected of excitons and biexcitons. Dominant excited-state resonances are ascribed to successive two-step absorptions of two photons of opposite optical orientation, one into an exciton excited state followed by another into a densely distributed biexciton excited state. Filling of the exciton state is found to lead to absorption from the exciton ground state to the biexciton excited state. Remaining excitation resonances with large oscillator strength are identified as two-photon absorption processes that directly create weakly correlated exciton pair states.