An analysis of the character of the possible dynamics of all hitherto known planetary satellites shows two satellites-Amalthea (J5) and Prometheus (S16)-to have the most unusual structure of the phase space of possible rotational motion. These are the only satellites whose phase space of planar rotation may host synchronous resonances of three different kinds: the α resonance, the β resonance, and a mode corresponding to the period doubling bifurcation of the α resonance. We analyze the stability of these states against the tilt of the rotational axis.