The atomic structure of the fivefold symmetric quasicrystal surface of Al70Pd21Mn9 was investigated by means of a dynamical low energy electron-diffraction analysis. Approximations are developed to address the high structural complexity of quasicrystals, yielding average layer-by-layer structural and compositional information. A mix of several relaxed bulklike terminations is found, consistently favoring dense Al-rich outermost atomic layers. These results can be understood in terms of principles known to govern surface structures of simpler, crystalline metals.