What's needed to put a PC fan into reverse?

Reversing the direction of rotation will be difficult. Since you only supply power, there is a controller in the fan that senses rotor position and commutates the motor accordingly. If you can get in there, you can probably reverse the direction by flipping two of the three sensors and two of the three windings. That won't be easy to do since these are all nicely integrated onto a small board.

Supplying negative power won't work, just fry the electronics.

However, you can still "reverse" the fan by simply installing it backwards.

Most PC fans are two phase with 4 pole stators, PM rotor and a single Hall sensor. I think I've seen 3-phase but not for more than 15 years.

Eg. (from here)

You may be able to just reverse the two coil wires to make this work. There is more than one way of ensuring that the two-phase motor (which otherwise would be happy to spin in either direction) starts in the correct direction.

It's also possible a small magnetic field would bias the sensor so it would spin in the opposite direction, but modern hall driver chips use edge detection and won't be fooled by that.

The circuit is a three phase inverter. Ultimately there will be three wires going from the PCB into the motor windings. Swap two of those three.