The pumping of interstellar methanol masers is studied, covering a range of excitation conditions. The Class I (Class A) masers result from collisional excitation followed by spontaneous radiative decay. Masers appear in low-frequency transitions where the upper state is strongly favoured by large Einstein A coefficients, associated with high-frequency transitions, or by slower loss rates. Absorption (anti-inversion) is seen when the lower state is so favoured. The Class II (Class B) masers appear when there is a source of continuum radiation warmer than the gas kinetic temperature. Here the high-frequency transitions selectively populate or depopulate levels by preferential absorption of submillimetre photons. This radiative pumping leads to just the opposite behaviour: masing in transitions which show absorption in the Class I case, and vice versa. In particular the very strong 12.2-GHz 20-3−1E and 6.6-GHz 51-60A+ masers are qualitatively accounted for, and predictions are made of other masing lines. This simple model accounts successfully for all the diverse methanol masers, switching between Class I and Class II behaviour, and explains naturally the phenomenon of lines which are seen as masing in some sources and in absorption in others; it also enables us to predict the occurrence of some additional methanol maser transitions not yet observed.