We use density-functional theory to study the ordered states of iridium manganese, demonstrating a simple but powerful method for describing magnetic structures and the transitions among them. As an illustrative example, the coupling of magnetism and crystal structure in IrMn is examined through a rigorous exploration of strain space. We then generate order parameters for the ground-state magnetic structures, providing a comprehensive framework for understanding antiferromagnetic variants and their magnetic anisotropy. In particular, we show how the most direct path between two variant structures can be analytically determined; this technique is used to equate two seemingly contradictory prior studies.