The input signal reference (positive and negative) may or may not be referenced the the supply. For some instrumentation amps, you can have the +/- pair be hundreds of volts away from the supply level.
Thus, for the use case you suggest, you have to check the data sheet to figure out how negative the input is allowed to go.
Another thing to do, if you don't need DC measurement, is to build a resistor bridge to get the "+" side centered, and then drive that "+" side from the input signal using a decoupling capacitor.