Semiconductor quantum dots, due to their small size, mark the transition between molecular and solid-state regimes, and are often described as `artificial atoms' (ref 1-3). This analogy originates from the early work on quantum confinement effects in semiconductor nanocrystals, where the electronic wavefunctions are predicted to exhibit atomic-like symmetries, for example `s ' and `p '. Spectroscopic studies of quantum dots have demonstrated discrete energy level structures and narrow transition linewidths, but the symmetry of the discrete states could be inferred only indirectly. Here we use cryogenic scanning tunnelling spectroscopy to identify directly atomic-like electronic states with s and p character in a series of indium arsenide nanocrystals. These states are manifest in tunnelling current-voltage measurements as two- and six-fold single-electron-charging multiplets respectively, and they follow an atom-like Aufbau principle of sequential energy level occupation.