Spin correlations and polarization in the top quark--antiquark system are measured using dilepton final states produced in $pp$ collisions at the LHC at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$. The data correspond to an integrated luminosity of $5.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected with the CMS detector. The measurements are performed using events with two oppositely charged leptons (electrons or muons), a significant imbalance in transverse momentum, and two or more jets, where at least one of the jets is identified as originating from a $b$ quark. The spin correlations and polarization are measured through asymmetries in angular distributions of the two selected leptons, unfolded to the parton level. All measurements are found to be in agreement with predictions of the standard model.