Complex oxides display highly tunable ground states, where small perturbations to the crystal structure can stabilize distinct electronic and magnetic phases. We use group theoretic methods and density functional theory calculations to analyze the polar crystal structure of A-site ordered SmBaMn₂O₆, which hosts a charge- and orbitally-ordered antiferromagnetic ground state. We show that a set of couplings between structural distortions are key for establishing the structural, electronic, and magnetic ground states and also establishes a hybrid improper ferroelectric mechanism in this material. We analyze the domain structure and show how the coupled degrees of freedom in SmBaMn₂O₆ leads to a network of coupled domains and domain wall vortices. Finally, we explore how competing electronic and magnetic phases may stabilize, for example at structural domain walls and in epitaxially strained thin films. This work provides new understanding of the complex physics realized across multiple length scales in SmBaMn₂O₆ and demonstrates a framework for the systematic exploration of correlated and structurally complex materials.