M. Orte-García, C. Esteban, J. García-Rojas, J. E. Méndez-Delgado, K. Z. Arellano-Córdova, A. Z. Lugo-Aranda, L. Toribio San Cipriano, F. F. Rosales-Ortega, I. R. Martínez-Hernández, E. Reyes-Rodríguez

Aims. We present a homogeneous observational study of electron temperature ($T_{\rm e}$) relations between ionic species: $T_{\rm e}$([N II]), $T_{\rm e}$([O II]), $T_{\rm e}$([O III]), $T_{\rm e}$([S II]), $T_{\rm e}$([S III]) and $T_{\rm e}$([Ar III]), using 699 spectra of Galactic and extragalactic H II regions and local star-forming galaxies (SFGs). Methods. We use the DEep Spectra of Ionised REgions Database Extended (DESIRED-E), comprising more than 3000 spectra with direct $T_{\rm e}$ determinations, selecting those with at least two $T_{\rm e}$ diagnostics. We recompute electron density ($n_{\rm e}$) and $T_{\rm e}$ using updated atomic data and a consistent methodology. The resulting $T_{\rm e}$--$T_{\rm e}$ relations are analysed using orthogonal distance regression, quantifying total and intrinsic dispersions and comparing slopes with previous works and photoionisation models. Results. Relations involving low-ionisation $T_{\rm e}$ diagnostics show large intrinsic dispersions, especially for $T_{\rm e}$([O II]) and $T_{\rm e}$([S II]), likely due to sensitivity to $n_{\rm e}$ inhomogeneities, recombination contributions, and uncertainties. In contrast, relations using $T_{\rm e}$([N II]) show lower dispersions, indicating that this diagnostic provides a more reliable estimate of the low-ionisation zone temperature when only higher-ionisation $T_{\rm e}$ diagnostics are available, despite observational difficulties at low metallicity. Overall, slopes agree with model predictions, particularly for relations with low intrinsic dispersion, such as those involving $T_{\rm e}$([N II]) and $T_{\rm e}$([S III]). These results provide a robust empirical basis for estimating $T_{\rm e}$ when limited diagnostics are available.