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