What does a Schmitt Trigger inverter act as when unpowered?
In theory, when a part is operated outside it working conditions (as documented in the datasheet), there are no guarantees on its behaviours. NONE.

In practice, most logic chip have input protection circuits that feed 'excess' input voltage to the power rail. So I would expect that an inverter with its input high has its output (somewhat - the drive characteritics will likely be far from optiomal) low. But with its input low there is no power, so the output could be either high-impedance or low or something inbetween.