Charged particle pitch angle distributions at one point on a magnetic drift surface in a rapidly rotating axisymmetric mirroring system (such as Jupiter's magnetosphere would be in the absence of the 10¿ dipole tilt) are related to those at another point by Liouville's theorem. If the distribution function in the rotating frame is gyrotropic, i.e., if it is independent of the phase angle of the gyration, it is gyrotropic at all points on that drift surface. Examples are given of 'pancake' and 'dumbbell' and isotropic distributions when they are observed from the nonrotating frame at different points on a drift surface.