A doublet formed by the coagulation of two colloidal spheres rotates toward alignment with an applied electric field. The rotation rate is proportional to the difference in the zeta potentials of the two spheres and a geometric coefficient N. This coefficient can be calculated from the electrohydrodynamic equations and depends on the kinematic boundary condition imposed on the spheres. If the doublet rotates as a single rigid body, so that the surfaces of the two spheres do not move relative to one another, then the value of N is a factor of 2 or 3 smaller than if the two spheres are free to rotate relative to one another. Thus, one can evaluate the rigidity of the doublet by determining the value of N experimentally. We have tracked the motion of individual doublets formed from micrometer-size polystyrene latex spheres differing in zeta potential and determined the value of N for different solution conditions. The influence of Brownian motion was taken into account when interpreting the rotational trajectories of the doublets. Over a range of electrolyte concentration, pH, sphere size, and electric field, all the values of N agree with the theoretical values for rigid-body rotation. This result is surprising because in many of the experiments the conditions were such that the two spheres should have been in a secondary minimum with a gap between 5 and 20 nm according to classical DLVO theory. In these cases the freely-rotating condition on each sphere is expected. Possible explanations for the rigid-body behavior of the doublets are suggested.