Several characterization results of a symmetric distribution based on concomitants of order statistics as well as k-records from Farlie-Gumbel-Morgenstern (FGM) family of bivariate distributions are established. These include characterizations of a symmetric distribution on the basis of equality in distribution, moments, Rényi and Tsallis entropies of concomitants of upper and lower order statistics, also in terms of the same properties of concomitants of upper and lower k-records,