My thinking on using a scope is to check the phase relationship between the two sets of windings. If on rises while the other falls against it they could theoretically cancel each other out. If they are in phase rising, together they should produce 220v when placed in series. Maybe I'm overthinking this.
Have you ever tried to run 2 120v things at the same time from the two outlets? That is one from each set of windings. My suggestion would be light bulbs(incandescent if you still got 'em) since they are cheap and easily replaced. If they both light then I think you only have one set of windings working and somehow connected to both sides of your breaker and if it is not too big a pain I would disconnect and separate the commons to verify with a ohm/continuity meter that there is actually two independent sets of windings. That is continuity from hot to common on each set and with the commons disconnected no continuity between the hots. Obviously engine not running. If I have made an error in my thinking someone please correct me.

And remember your volt meter reads voltage drop between the test points not voltage present.