The method of "rotational electrophoresis" has previously been used to measure charge nonuniformity on colloidal particles. In this technique, one measures the angular velocities of many particles in electrophoresis and interprets these angular velocities in terms of electrokinetic theory, assuming a random distribution of charge on the individual particles. In the present paper, we verify whether the charge distribution is in fact statistically random by examining the stationary angle to which spheroidal particles (or similar particles, such as doublets) rotate. Calculations are provided for the average stationary angle to which spheroids (axis ratio of 0.1-10) rotate. Estimates are given for how strongly the spheroids remain at a steady angle despite Brownian motion. In addition, experimental data are reported for polystyrene latex particles, showing that the charge distribution on the particles is random for about half of the particles.