The X-ray polarization anisotropy of anomalous scattering in crystals of brominated nucleic acids and selenated proteins is shown to have significant effects on the diffraction data collected at an absorption edge. For conventionally collected single- or multi-wavelength anomalous diffraction data, the main manifestation of the anisotropy of anomalous scattering is the breakage of the equivalence between symmetry-related reflections, inducing intensity differences between them that can be exploited to yield extra phase information in the structure-solution process. A new formalism for describing the anisotropy of anomalous scattering which allows these effects to be incorporated into the general scheme of experimental phasing methods using an extended Harker construction is introduced. This requires a paradigm shift in the data-processing strategy, since the usual separation of the data-merging and phasing steps is abandoned. The data are kept unmerged down to the Harker construction, where the symmetry-breaking is explicitly modelled and refined and becomes a source of supplementary phase information. These ideas have been implemented in the phasing program SHARP. Refinements using actual data show that exploitation of the anisotropy of anomalous scattering can deliver substantial extra phasing power compared with conventional approaches using the same raw data. Examples are given that show improvements in the phases which are typically of the same order of magnitude as those obtained in a conventional approach by adding a second-wavelength data set to a SAD experiment. It is argued that such gains, which come essentially for free, i.e. without the collection of new data, are highly significant, since radiation damage can frequently preclude the collection of a second-wavelength data set. Finally, further developments in synchrotron instrumentation and in the design of data-collection strategies that could help to maximize these gains are outlined.