We study the production of Mueller-Navelet jets at hadron colliders in the Balitsky-Fadin-Kuraev-Lipatov (BFKL) framework. We show that a measurement of the relative azimuthal angle DeltaPhi between the jets can provide a good testing ground for corrections due to next-leading logarithms (NLL). Besides the well-known azimuthal decorrelation with increasing rapidity interval Deltaeta between the jets, we propose to also measure this effect as a function of R=k_2/k_1, the ratio between the jets transverse momenta. Using renormalisation-group improved NLL kernel, we obtain predictions for dsigma/dDeltaeta dR dDeltaPhi. We analyse NLL-scheme and renormalisation-scale uncertainties, and energy-momentum conservation effects, in order to motivate a measurement at the Tevatron and the LHC.