We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS (input-to-state stability) framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability - characterizations of the property as conjunctions of weaker stability notions. When applied to the notion of integral ISS, these characterizations yield analogous results.