Pitch stability is dependent on the rate of change of pitching moment with lift, not the actual value of the pitching moment.
The pitching moment requirement for an airfoil for a flying wing is that it be zero or slightly positive. It it isn't, negative elevator deflection is required to get trim. This results in the reflexed camber line you refer to. The NACA 1R2 and 2R2 families of airfoils have such a reflexed camber line.