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