Well, that makes a comparator. Note that the input signal is ground-referenced (assumed), so the amp can only compare its value to ground, and draw either short-circuit current from the respective pin, or saturate down to V_OH/V_OL as the case may be. Note that only one or the other supply will be active at a time, so the bypass cap isn't helpful here. Probably.
This is where we need to know the true nature of an opamp.
Most have a class AB to B output stage, meaning that quiescent current flows + to -, and that current doesn't change much, while the output current adds on top of it, in the respective direction (+V to OUT / OUT to -V).
Some have a fair amount of class A; whereas LM358 is class B (in fact having noticeable crossover distortion even), NE5534 is notoriously deep class A (to the extent that, operated at rated supply voltage, the chips are prone to overheating!). In class A, we expect the balance of quiescent current to shift, at least until one side or the other enters cutoff, and then the above will be true.
So it is, with the "probably" I left above. In class A, the supply current remains constant and the balance shifts, thus the bypass capacitor sees no current flow (not to say it's irrelevant, more that it's able to do its job, so to speak) and the outputs shift in kind. In class AB, only one side moves inward depending on input polarity, and the capacitor will actually act to pull the opposite side opposite its active direction! This may be an unintended effect of your scheme -- but it may be quite intentionally used, as a kind of absolute-value function, say? (Once stabilized with feedback from the circuit output, of course!)
As it happens, a similar sort of topology is even used to model the supply currents in many op-amp simulation (SPICE) models themselves. I'm... not sure that there's much documentation about this out there; and, hrm, I've worked on such models before, but I don't seem to have any screenshots handy, unfortunately. So, I'll leave this as flavor, I guess.
Tim