If an axially symmetric stationary metric is given in standard form (i.e. in coordinates adapted to the symmetries) the transformation =-t (=constant) of the azimuthal angle leads to another such standard form. The spatial latticeL corresponding to the latter rotates at angular velocity relative to the latticeL of the former. For the standard form of a stationary metric there are simple formulae giving the four-acceleration of a given lattice point and the rotation of a gyroscope at a given lattice point. Applying these formulae toL, we find the condition for circular paths about the axis inL to be 4-geodesic, and also the precession of gyroscopes along circular paths which are not necessarily geodesic. Among other examples we re-obtain the complete geodesic structure of the Gödel universe, and the gyroscopic precessions associated with the names of Thomas, Fokker and de Sitter, and Schiff.