Rotation Sensitivity in Oriented Detectors Is Not One Quantity

Oriented object detectors can be accurate without being equivariant to rotations of their input, and this lack of equivariance is usually summarised by a single number. We split the AP cost of the four axis-aligned views into a quarter-turn mode C, a half-turn mode H and a signed chirality Q, three fixed combinations of the discrete Fourier coefficients of per-view AP, and we measure the disagreement between predictions on different views separately from its AP cost. Both measurements need an exact map back to the original frame. Mapping predictions back with the pixel-index form of the inverse rotation is off by one pixel for continuous box coordinates, displaces the half-turn view along both axes and inflates H, so we check the mapping against the array rotation it inverts. In a 2×2 augmentation factorial on DOTA with paired seeds, each mode is removed by the augmentation that covers the rotations it measures. Adding the vertical flip, whose composition with the horizontal flip is the half turn, removes H (macro AP50 0.0218 → 0.0005) but leaves most of C (0.0505 → 0.0418), whereas adding rotation removes both (C = −0.0049, H = −0.0015). Bootstrap intervals over seeds and scenes separate H from zero only in cells trained without the half turn, and the pattern holds at a second model capacity, a second input size and a stricter overlap threshold. A control with rotations of at most 15° improves accuracy but removes neither mode. After rotation training no rotated view has a detectable AP cost, although, averaged over classes, more than a tenth of the predictions have no counterpart when the canonical and a rotated view are matched in either direction. A detector with a rotation-equivariant backbone and the same flips keeps about a third of the C that flips alone leave in our detectors (0.0132 against 0.0418). Because a half turn leaves box orientations unchanged, an explanation based on box orientation can only account for C. On DOTA, classes that are more anisotropic under a quarter turn lose more C under rotation training (Spearman ρ = −0.62, permutation p = 0.018), and H shows no such relation. This class-level relation does not hold on SODA-A, where C is indistinguishable from zero. Some small residuals remain, including a half-turn cost in the equivariant detector and a negative chirality when rotation is added without the vertical flip, and we report the tests we ran on them.