Quantification of the uncertainty of distribution functions, by using entropy and extropy, is important in many statistical analyses. Inspired by this, our study uses extropy and several related measures (including the cumulative residual extropy, cumulative past extropy, and extropy-inaccuracy measure) of order statistics (OSs) to offer multiple characterizations of symmetric continuous distributions. We demonstrate that a defining feature of symmetric distributions is the equality of these measures of upper and lower OSs. Using concomitants of OSs based on the bivariate distributions belonging to the Farlie-Gumbel-Morgenstern (FGM) family, the same characteristic is demonstrated for these measures. Finally, a real data set is used to illustrate the applicability of the suggested test.