Normalized double-differential cross sections for top quark pair (t t ¯) production are measured in pp collisions at a centre-of-mass energy of 8TeV with the CMS experiment at the LHC. The analyzed data correspond to an integrated luminosity of 19.7fb-1. The measurement is performed in the dilepton e ±μ∓ final state. The t t ¯ cross section is determined as a function of various pairs of observables characterizing the kinematics of the top quark and t t ¯ system. The data are compared to calculations using perturbative quantum chromodynamics at next-to-leading and approximate next-to-next-to-leading orders. They are also compared to predictions of Monte Carlo event generators that complement fixed-order computations with parton showers, hadronization, and multiple-parton interactions. Overall agreement is observed with the predictions, which is improved when the latest global sets of proton parton distribution functions are used. The inclusion of the measured t t ¯ cross sections in a fit of parametrized parton distribution functions is shown to have significant impact on the gluon distribution.