Rotary analysis decomposes vector motions on the plane into counter-rotating components, which have proved particularly useful in the study of geophysical flows influenced by the rotation of the Earth. For stationary random signals, the motion at any frequency takes the form of a random ellipse. Although there are numerous applications of rotary analysis, relatively little attention has been paid to the statistical properties of the random ellipses or to the estimated rotary coefficient, which measures the tendency to rotate counterclockwise or clockwise. The precise statistical structure of the ellipses is reviewed, including the random behaviour of the ellipse orientation, aspect ratio and intensity. Special attention is then paid to spectral matrix estimation from physical data and to hypothesis testing and confidence intervals computed using the estimated matrices.