At geomagnetic field strengths, polymer organic light-emitting diodes (OLEDs) exhibit a giant anisotropy in magnetoresistance of up to 35%. Comparison of the effect arising from a protonated and an equivalent perdeuterated conjugated polymer, in combination with semiclassical quantum-stochastic modeling, demonstrates that microscopically anisotropic hyperfine-field distributions, on the level of the individual molecules, constitute the primary cause for this effect. For this microscopic anisotropy to emerge in the ensemble there must also be some degree of macroscopic ordering, which may arise from the structural anisotropy of the polymer. The theory predicts a critical field range, for which the anisotropy transitions from a twofold to a fourfold and back to a twofold angular functionality with increasing field strength, over a field range of only a few microtesla. Such a transition is indeed found experimentally, although it spans over a somewhat larger field range, suggesting a level of material disorder that is not accounted for in the simulations. Through the combination with microscopic modeling, anisotropic magnetoresistance can serve as a sensitive probe of macroscopic molecular ordering in organic semiconductors. The inclination compass effect in OLEDs also offers a potential route for probing the radical-pair mechanism of spin chemistry in the solid state, and the associated coherent and incoherent electronic and nuclear spin dynamics at room temperature, and could point to an intriguing analogy to some forms of avian magnetoreception.