The original inverter board actually runs off a 16v source from the main chassis if it's all original. But will also run from a 12v source if using a switching power supply.
The ground is accomplished through the video signal input ground connection.... there is no other ground wire for the power. Just a single positive voltage lead.
If you are running an original Nintendo board, then you do not need power to the inverter board. (heck, you don't need the board at all actually)
If you are running a JAMMA type board then you DO need power to it otherwise it will not work.
One of them kicks out normal video, the other inverted video.
Do you mean that the picture is upside down/backwards?
If so, then it has nothing to do with that "inverter" board...... the original inverter board deals with the color inversion, not the physical picture orientation.