A combination of measurements of the inclusive top-quark pair production cross-section performed by ATLAS and CMS in proton--proton collisions at centre-of-mass energies of 7 and 8 TeV at the LHC is presented. The cross-sections are obtained using top-quark pair decays with an opposite-charge electron-muon pair in the final state and with data corresponding to an integrated luminosity of about 5 inverse fb at 7 TeV and about 20 inverse fb at 8 TeV for each experiment. The combined cross-sections are determined to be 178.5 +- 4.7 pb at 7 TeV and 243 +6.0 -5.9 pb at 8 TeV with a correlation of 0.41 for a reference top-quark mass value of 172.5 GeV. The ratio of the combined cross-sections is determined as R_8/7 = 1.363 +- 0.032. The combined measured cross-sections and their ratio agree well with theory calculations using several parton distribution function (PDF) sets. The values of the top-quark pole mass (with the strong-coupling constant fixed at 0.118) and the strong-coupling constant (with the top-quark pole mass fixed at 172.5 GeV) are extracted from the combined results by fitting a next-to-next-to leading order plus next-to-next-to leading log QCD prediction to the measurements, using the NNPDF31_nnlo_ascorr_notop PDF set, yielding mtpole = 173.4 + 1.8 - 2.0 GeV and alpha_s = 0.1170 +- 0.0021 respectively