Precision measurements are reported of the cross-spectrum of rotationally induced differential position displacements in a pair of colocated 39 m long, high-power Michelson interferometers. One arm of each interferometer is bent 90° near its midpoint to obtain sensitivity to rotations about an axis normal to the plane of the instrument. The instrument achieves quantum-limited sensing of spatially correlated signals in a broad frequency band extending beyond the 3.9-MHz inverse light travel time of the apparatus. For stationary signals with bandwidth Δf>10 kHz, the sensitivity to rotation-induced strain h of classical or exotic origin surpasses CSD_{δh}<t_{P}/2, where t_{P}=5.39×10^{-44} s is the Planck time. This measurement is used to constrain a semiclassical model of nonlocally coherent rotational degrees of freedom of spacetime, which have been conjectured to emerge in holographic quantum geometry but are not present in a classical metric.