The crystal structure of NaNbO3 has been studied in detail in the temperature regime 360 < T < 520 degrees C using a combination of high-resolution neutron and synchrotron X-ray powder diffraction, supported by first-principles calculations. A systematic symmetry-mode analysis is used to determine the presence of the key active distortion modes that, in turn, provides a small and an unambiguous set of trial structural models. A unique model for Phase S (480 < T < 510 degrees C) is elucidated, having a 2 x 2 x 4 superlattice of the aristotype perovskite structure, space group Pmmn. This unusual and unique structure features a novel example of a compound octahedral tilt system in a perovskite. Two possible structural models for Phase R (370 < T < 470 degrees C) are determined, each having a 2 x 2 x 6 superlattice and differing only in the nature of the complex tilt system along the 'long' axis. It is impossible to identify a definitive model from the present study, although reasons for preferring one over the other are discussed. Some of the possible pitfalls in determining such complex, pseudosymmetric crystal structures from powder diffraction data are also highlighted.