Context. Accurate determinations of chemical abundances in H II regions and star-forming galaxies rely critically on robust measurements of the electron temperature (Te) across different ionic species. However, when using a two- or three-zone scheme to approximate the ionisation structure, it is usually possible to measure only a single temperature diagnostic, which makes it necessary to rely on empirical or theoretical relations to infer the temperatures in the remaining zones. Aims. We present a homogeneous observational study of electron temperature (Te) relations between different ionic species: Te([N II]), Te([O II]), Te([O III]), Te([S II]), Te([S III]), and Te([Ar III]), using a sample of 699 spectra of Galactic and extragalactic HII regions and star-forming galaxies (SFGs) of the local Universe. Methods. We used the DEep Spectra of Ionised REgions Database Extended project (DESIRED-E) that comprises more than 3000 spectra of H II regions and SFGs with direct determinations of Te. From this database, we selected those spectra where it is possible to determine Te for at least two ionic species. We recalculated the electron density (ne) and Te using updated atomic data and a fully consistent methodology. The resulting Te-Te relations were examined using orthogonal distance regression fits, with their total and intrinsic dispersions quantified, and with the fitted slopes being compared with results from previous works and predictions from photoionisation models. Results. Relations involving low-ionisation Te diagnostics show large intrinsic dispersions, particularly those based on Te([O II]) and Te([S II]), which is likely due to their higher sensitivity to ne inhomogeneities, possible recombination contributions, and observational uncertainties. Relations using Te([N II]) exhibit systematically lower dispersions, indicating that this diagnostic provides a more reliable estimate of the low-ionisation zone temperature when only higher-ionisation diagnostics are available, despite its observational difficulty at a low metallicity. Overall, the fitted slopes are broadly consistent with photoionisation model predictions, particularly for relations with low intrinsic dispersion, such as those involving Te([N II]) and Te([S III]). This work provides the most comprehensive observational characterisation to date of Te-Te relations in local H II regions. The empirical relations presented here offer a robust basis for estimating Te, especially when only a single diagnostic is available.