The electron temperatures of high-metallicity [12 + log(O/H) > 8.2] HII regions have been studied. The empirical flux-flux relations which express the nebular-to-auroral [OIII] line ratio Q3,O (as well as the nebular-to-auroral [OII] line ratio Q2,O, and the nebular-to-auroral [NII] line ratio Q2,N) in terms of the nebular R3 and R2 line fluxes in spectra of high-metallicity HII regions are derived, and the electron temperatures t3,O, t2,O and t2,N in a number of extragalactic HII regions are also determined. Furthermore, the t2- t3 diagram is discussed. It is found that there is a one-to-one correspondence between t2 and t3 electron temperatures for HII regions with a weak nebular R3 lines (logR3 <~ 0.5). The derived t2,N-t3,O relation for these HII regions is similar to commonly used t2-t3 relations. The HII regions with a strong nebular R3 line flux (logR3 >~ 0.5) do not follow this relation. A discrepancy between t2,N and t2,O temperatures is found, being the t2,N temperatures systematically lower than the t2,O ones. The differences are small at low electron temperatures, and increase with increasing electron temperatures up to 10 per cent at t = 1. The uncertainties in the atomic data may be the cause of this discrepancy.