Long slit spectrophotometry of 13 bipolar planetary nebulae has been used to study their physical and chemical properties. In each nebula, one to seven different spatial regions have been considered in order to search for possible abundance variations through the nebulae. We obtained the following main results: - within the errors, the He, O, and N abundances are constant through all the nebulae. - the Ne, Ar, and S abundances are also constant, within the errors, but their face values have systematic increases toward the outer regions of the nebulae. These trends may be attributed to inaccuracies in the ionization correction factors, as predicted by Alexander & Balick (1997) for long-slit observations of extended PNe. The corresponding increase of the N abundance predicted by those authors is, however, generally not observed (with one exception). - The present sample contains some of the Galactic PNe with the highest He and N/O abundances known to date (M 3-2, He 2-111, NGC 6537). The highest He overabundances cannot by reproduced by any current model of AGB evolution. - Oxygen depletion is suggested for the nebulae with the highest N/O abundances, indicating that efficient ON cycle process has occurred in their progenitors. Tables 4 to 29 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via /ttp://cdsweb.u-strasbg.fr/Abstract.html}. Based on observations made at the European Southern Observatory