Normalized double-differential cross sections for top quark pair (t (t) over bar) production are measured in pp collisions at a centre-of-mass energy of 8 TeV with the CMS experiment at the LHC. The analyzed data correspond to an integrated luminosity of 19.7 fb(-1). The measurement is performed in the dilepton e(+/-)mu(+/-) final state. The tt cross section is determined as a function of various pairs of observables characterizing the kinematics of the top quark and tt 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 tt cross sections in a fit of parametrized parton distribution functions is shown to have significant impact on the gluon distribution.