  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 $90^{\circ}$ 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 $\Delta f > 10\;\mathrm{kHz}$, the
sensitivity to rotation-induced strain $h$ of classical or exotic origin
surpasses $\mathrm{CSD}_{\delta h} < t_P / 2$, where $t_P = 5.39 \times
10^{-44}\;\mathrm{s}$ 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.
