The National Technology Roadmap for Semiconductors has set a goal of 0.1-0.07 mu m features for the end of the decade. Improved calibration methods will be required to achieve this goal. A rigorous method for using an electron (e)-beam stage to calibrate itself, known as stage self-calibration, was published in 1985 and has since been refined in various ways and shown to be capable of accuracy throughout the workspace in the range of the machine＇s repeatability. This article presents symmetry and invariant sets as the governing features of self-calibration, using classical reversal methods as illustrations. The term symmetry is used in a dual sense; it refers both to the symmetry of a pattern of points and to the rigid motions that characterize the symmetry of the pattern. The geometric perspective gained in this way can be exploited to derive techniques and explicit error bounds for high-accuracy self-calibration, The article will be of interest to process engineers and metrologists seeking insights and methods for improving the accuracy of high-precision stages.