Iridium manganese (IrMn) is arguably the most important antiferromagnetic material for device applications due to its metallic nature, high Néel temperature, and exceptionally high magnetocrystalline anisotropy. Despite its importance, its magnetic properties are poorly understood due to its intrinsic complexity and the interplay between structural and magnetic properties. Here we present a unifying atomistic model of IrxMn(1-x) alloys which reproduces the key experimental facts of the material, while providing unprecedented understanding of the compositional and structural origins of its magnetic ground state and thermodynamic properties. We find that the Néel temperature is strongly dependent on the nature of the ground-state magnetic order which varies with x from a triangular to tetrahedral spin structure, leading to different levels of geometric spin frustration. The Néel temperature increases linearly with manganese concentration for the disordered phase, while the ordered phases show a peak for Ir50Mn50 followed by a decrease due to increased spin frustration. The ground-state tetrahedral spin structure of the disordered phase is composition independent for manganese concentrations in the 50-95% range, while the degree of spin order varies strongly in the same range. For low manganese concentrations, we find antiferromagnetic spin-glass and ferromagnetic ground-state spin structures. The magnetic anisotropy energy exhibits a complex dependence on the lattice symmetry, presenting easy-plane, cubic, and unconventional symmetries for the principal phases, and a similarly complex variation of magnitude. The complexity of behavior represents a dual blessing and a curse in that the properties of a particular sample depend strongly on the degree of order and composition, while also providing a large state space to engineer an antiferromagnet with optimal symmetry, magnetic anisotropy, and thermal stability. Such effects are important for the future development of nanoscale sensor devices and antiferromagnetic spintronics.