Optical and electronic phenomena in solids arise from the behaviour of electrons and holes (unoccupied states in a filled electron sea). Electron-hole symmetry can often be invoked as a simplifying description, which states that electrons with energy above the Fermi sea behave the same as holes below the Fermi energy. In semiconductors, however, electron-hole symmetry is generally absent, because the energy-band structure of the conduction band differs from the valence band. Here we report on measurements of the discrete, quantized-energy spectrum of electrons and holes in a semiconducting carbon nanotube. By applying a voltage to a gate electrode, an individual nanotube is filled controllably with a precise number of either electrons or holes, starting from one. The discrete excitation spectrum for a nanotube with N holes is strikingly similar to the corresponding spectrum for N electrons. This observation of near-perfect electron-hole symmetry demonstrates that a semiconducting nanotube can be free of charged impurities, even in the limit of few electrons or holes. We furthermore find an anomalously small Zeeman spin splitting and an excitation spectrum indicating strong electron-electron interactions.