[wilkin250r]

The single-wire approach is fairly simple in concept, but slightly more difficult to put into practice.

In theory, a simple buck/boost converter module to produce a -12V from a +12V power, then a relay to switch your single wire between +12V and -12V would essentially be the same as reversing polarity.

I can find pre-made DC-DC converter modules easily, many less than $5, and not much bigger than a relay. But to work, it either needs to offer full isolation, or it needs to specifically invert the source voltage. These aspects are simple enough design if I was doing the circuit from scratch, but very rare to find in a pre-made module. I've found a couple in the realm of $100, but nothing on the realm of $8.