The well-known noncommutativity of three-dimensional finite rotations has long been a curiosity in mechanics since, in actual solution of dynamical problems, the angular velocity, which is conveniently representable as a vector, plays a more natural role. In modern inertial guidance systems, however, the orientation of a body in space, i.e., a rotation, is of primary engineering interest. In this paper a simple method of determining orientation from the time history of three body components of angular velocity is developed by means of a new theorem in kinematics. As a special case of this theorem it is shown that a gyro subjected to a regime of rotations which returns it to the original space orientation will, in general, produce a residual signal. It will have experienced a nonzero and easily calculated mean angular velocity about its input axis. Some implications of the theorem for the design of inertial guidance systems and for the testing of gyros are discussed.