Although others that have responded to this thread have far more experience, I believe Peltiers are optimized to work in one direction. That is to say there is a hot and cold side, so I'm not sure it is as simple as adding an inverter. Not that you'd likely be using your hands as a heat sync anyway.

Yes, I realize this was meant as a joke, but I'm still curious if current can be reversed by just changing the gradient. Does it still work but maybe it is less efficient? Is there no difference at all?

Ah, good point. I hadn't considered that. It looks like you are correct! You might sandwich two Seebeck-Peltier devices and switch between them, though.

https://en.wikipedia.org/wiki/Thermoelectric_effect