If the inspector was right, what is wrong with the wiring and how do I
fix it?

Swap black and white at both ends. This involves working on your breaker
panel, which is dangerous. I recommend hiring an electrician.

If the inspector was wrong, how could that be, as he was right about
two other plugs in the same room?

He'll only be wrong if he mixed up his results after making measurements.

Or do I need to spend a couple of bucks on a sensor to check for
myself whether the inspector was right?

That's what I'd do. And don't let the erroneous situation persist.