Attached is a screenshot of what I have.

I want +6 / -6V so that the voltage across the inductor is centered at zero. I think this is how induction heaters would work. Or do I even need this?

As you can see in the screenshot, the switched current isn't going going into the negatives. It's staying positive. I thought it should be symmetric across zero.

 

The usual way would be to use an H-bridge (4 MOSFETs) and a single-polarity supply.

 

Hello,

There are a lot of schematic on google for induction heaters:
induction heater schematic diagram

Keep in mind that you will need quite some power to heat something.

Bertus

 

As you can see in the screenshot, the switched current isn't going going into the negatives. It's staying positive. I thought it should be symmetric across zero.

The screenshot shows a chart ending at 13.5ms without achieving steady state. Try letting it run longer.