We use a sample of 154 observations of 124 HII regions that have measurements of both T_e_[OIII] and T_e_[NII], compiled from the literature, to explore the behaviour of the T_e_[OIII]-T_e_[NII] temperature relation. We confirm that the relation depends on the degree of ionization and present a new set of relations for two different ranges of this parameter. We study the effects introduced by our temperature relations and four other available relations in the calculation of oxygen and nitrogen abundances. We find that our relations improve slightly on the results obtained with the previous ones. We also use a sample of 26 deep, high-resolution spectra to estimate the contribution of blending to the intensity of the temperature-sensitive line [OIII]{lambda}4363, and we derive a relation to correct T_e_[OIII] for this effect. With our sample of 154 spectra, we analyse the reliability of the R, S, O3N2, N2, ONS, and C strong-line methods by comparing the metallicity obtained with these methods with the one implied by the direct method. We find that the strong-line methods introduce differences that reach ~0.2dex or more, and that these differences depend on O/H, N/O, and the degree of ionization.