We present a simultaneous measurement of the forward-backward asymmetry and the top-quark polarization in tt¯ production in dilepton final states using 9.7  fb−1 of proton-antiproton collisions at s√=1.96  TeV with the D0 detector. To reconstruct the distributions of kinematic observables we employ a matrix element technique that calculates the likelihood of the possible tt¯ kinematic configurations. After accounting for the presence of background events and for calibration effects, we obtain a forward-backward asymmetry of Att¯=(15.0±6.4(stat)±4.9(syst))% and a top-quark polarization times spin analyzing power in the beam basis of κP=(7.2±10.5(stat)±4.2(syst))%, with a correlation of −56% between the measurements. If we constrain the forward-backward asymmetry to its expected standard model value, we obtain a measurement of the top polarization of κP=(11.3±9.1(stat)±1.9(syst))%. If we constrain the top polarization to its expected standard model value, we measure a forward-backward asymmetry of Att¯=(17.5±5.6(stat)±3.1(syst))%. A combination with the D0 Att¯ measurement in the lepton+jets final state yields an asymmetry of Att¯=(11.8±2.5(stat)±1.3(syst))%. Within their respective uncertainties, all these results are consistent with the standard model expectations.