In this paper the relationships between stability at the input and at the output of a linear non-unilateral 2-port are analyzed. For this purpose the existence of a duality mapping between the two ports is shown and, by using the main properties of Möbius transforms, new mutual relationships between the stability conditions at input and output ports are demonstrated. Such relationships add to the case of unconditional stability for which it is well known that unconditional stability at the input implies unconditional stability at the output (and vice versa). This concept will be extended to all the possible cases of reciprocal position between the stability area in the load reflection coefficient plane and the Smith circle, showing that, for a given situation at the output, only one corresponding situation can be observed at the input (and vice versa). Limit cases are further considered.