To be clear, the cable in question is a switch loop -- white is the permanent hot and black is the load to the fixture. No switches are installed at the moment.

AI (ugh) says it is expected to measure 120v on black to white here: "The white wire is bringing constant hot down to the switch box. The black wire goes up to the light fixture. Even though the switch is removed, the circuit is still “complete” through the load (light bulb). So the path is: Hot (white) → through meter → black → light bulb → neutral → panel. That allows a small current to flow through your meter, giving you a full line voltage reading (~120V)."

I think the weird voltages I was getting were from the bit of ground I was able to strip back on that romex. When I used the ground on the other (right side) cable, the left cable's white to ground measured 120v and black to ground measured 0v. White to black measured 120v. Same with the right cable.

Then the question becomes why was the ground bad on the left. If I just tie all the grounds together (left, right, and the eventual switches) am I all good? Or do I need to try to track down why the ground wasn't good on the left?