Orbital precession changes the seasonal distribution of insolation at a given latitude but not the annual mean. Hence, the correlation of paleoclimate proxies of annual-mean precipitation with orbital precession implies a nonlinear rectification in the precipitation response to seasonal solar forcing. It has previously been suggested that the relevant nonlinearity is that of the Clausius-Clapeyron relationship. Here it is argued that a different nonlinearity related to moisture advection by the atmospheric circulation is more important. When perihelion changes from one hemisphere's summer solstice to the other's in an idealized aquaplanet atmospheric general circulation model, annual-mean precipitation increases in the hemisphere with the brighter, warmer summer and decreases in the other hemisphere, in qualitative agreement with paleoclimate proxies that indicate such hemispherically antisymmetric climate variations. The rectification mechanism that gives rise to the precipitation changes is identified by decomposing the perturbation water vapor budget into "thermodynamic" and "dynamic" components. Thermodynamic changes (caused by changes in humidity with unchanged winds) dominate the hemispherically antisymmetric annual-mean precipitation response to precession in the absence of land-sea contrasts. The nonlinearity that enables the thermodynamic changes to affect annual-mean precipitation is a nonlinearity of moisture advection that arises because precession-induced seasonal humidity changes correlate with the seasonal cycle in low-level convergence. This interpretation is confirmed using simulations in which the Clausius-Clapeyron relationship is explicitly linearized. The thermodynamic mechanism also operates in simulations with an idealized representation of land, although in these simulations the dynamic component of the precipitation changes is also important, adding to the thermodynamic precipitation changes in some latitudes and offsetting it in others.