We consider synthesis and analysis of probability measures using the entropy-regularized Wasserstein-2 cost and its unbiased version, the Sinkhorn divergence. The synthesis problem consists of computing the barycenter, with respect to these costs, of reference measures given a set of coefficients belonging to the simplex. The analysis problem consists of finding the coefficients for the closest barycenter in the Wasserstein-2 distance to a given measure. Under the weakest assumptions on the measures thus far in the literature, we compute the derivative of the entropy-regularized Wasserstein-2 cost. We leverage this to establish a characterization of barycenters with respect to the entropy-regularized Wasserstein-2 cost as solutions that correspond to a fixed point of an average of the entropy-regularized displacement maps. This characterization yields a finite-dimensional, convex, quadratic program for solving the analysis problem when the measure being analyzed is a barycenter with respect to the entropy-regularized Wasserstein-2 cost. We show that these coefficients, as well as the value of the barycenter functional, can be estimated from samples with dimension-independent rates of convergence, and that barycentric coefficients are stable with respect to perturbations in the Wasserstein-2 metric. We employ the barycentric coefficients as features for classification of corrupted point cloud data, and show that compared to neural network baselines, our approach is more efficient in small training data regimes.