The nonlinear Poisson-Boltzmann theory is used to calculate the electrical double-layer force and interaction free energy between dissimilarly charged surfaces. For symmetric electrolytes, in addition to cases in which both surfaces maintain constant surface potential or constant surface charge during interaction, we also consider cases in which one surface maintains constant surface potential and the other maintains constant surface charge. We further present a general algorithm for calculating the double layer force and interaction free energy between surfaces with ionizable surface groups across electrolytes of any valence or composition. These results suggest interesting features of the double-layer interaction that can be observed by direct force measurement techniques.