The task of mapping two or more distributions to a shared representation has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. However, most existing formulations only consider the setting of two distributions, and moreover, do not have an identifiable, unique shared latent representation. We use optimal transport theory to consider a natural multiple distribution extension of the Monge assignment problem we call the symmetric Monge map problem and show that it is equivalent to the Wasserstein barycenter problem. Yet, the maps to the barycenter are challenging to estimate. Prior methods often ignore transportation cost, rely on adversarial methods, or only work for discrete distributions. Therefore, our goal is to estimate invertible maps between two or more distributions and their corresponding barycenter via a simple iterative flow method. Our method decouples each iteration into two subproblems: 1) estimate simple distributions and 2) estimate the invertible maps to the barycenter via known closed-form OT results. Our empirical results give evidence that this iterative algorithm approximates the maps to the barycenter.