A single-crystal charge-flipping algorithm has been applied to two-dimensional projections derived from X-ray powder diffraction data to retrieve structure-factor phases. These phases proved to be as reliable as those obtained from high-resolution transmission electron microscopy (HRTEM) images or from precession electron diffraction data. In particular, the stronger reflections tend to be correctly phased. The two-dimensional electron-density `images' obtained in this way show the same features as the corresponding HRTEM images, but with higher resolution. Application of the powder charge-flipping algorithm to the full three-dimensional powder diffraction data in conjunction with phases derived from several such (arbitrarily selected) projections was found to have a significant and beneficial effect on the structure solution. The approach was first developed using data collected on the complex zeolite TNU-9, and was then tested further using data for IM-5 and SSZ-74, two similarly complex zeolites. All three of these structures were originally solved by combining X-ray powder diffraction and electron microscopy data, because X-ray diffraction data alone were not sufficient. In all three cases, the phase information derived from two-dimensional subsets of the X-ray powder diffraction data resulted in a significant improvement in the electron-density maps generated by the powder charge-flipping algorithm. The inclusion of this phase information allowed all three structures to be determined from the X-ray data alone. This two-dimensional X-ray powder diffraction approach appears to offer a remarkably simple and powerful method for solving the structures of complex polycrystalline materials.