We present deep high-resolution (R ˜ 15 000) and high-quality UVES optical spectrophotometry of nine planetary nebulae with dual-dust chemistry. We compute physical conditions from several diagnostics. Ionic abundances for a large number of ions of N, O, Ne, S, Cl, Ar, K, Fe and Kr are derived from collisionally excited lines. Elemental abundances are computed using state-of-the-art ionization correction factors. We derive accurate C/O ratios from optical recombination lines. We have re-analysed additional high-quality spectra of 14 PNe from the literature following the same methodology. Comparison with asymptotic giant branch models reveals that about half of the total sample objects are consistent with being descendants of low-mass progenitor stars (M < 1.5 M⊙). Given the observed N/O, C/O and He/H ratios, we cannot discard that some of the objects come from more massive progenitor stars (M > 3-4 M⊙) that have suffered a mild hot bottom burning. None of the objects seem to be descendant of very massive progenitors. We propose that in most of the planetary nebulae studied here, the polycyclic aromatic hydrocarbons have been formed through the dissociation of the CO molecule. The hypothesis of a last thermal pulse that turns O-rich PNe into C-rich PNe is discarded, except in three objects, that show C/O > 1. We also discuss the possibility of an He pre-enrichment to explain the most He-enriched objects. We cannot discard another scenarios like extra mixing, stellar rotation or binary interactions to explain the chemical abundances behaviour observed in our sample.