Each phase is made of two coils that can be connected in series or parallel, like a dual-voltage transformer, right?
If you energize each polarity-known coil with, say, a 12v transformer, and place the polarity-unknown coil in series with it (like an auto-transformer), couldn't the result, either 24v or 0v, tell you the polarity of the unknown coil?
Larry, the math does not work out perfectly - but what you suggest is spot on.
One reason it does not work out perfect is that you do not have access to the mid point of the inner Y. It's a 9 lead motor and not 12, so they don't bring that inner connection out. So if you energize, for example, leads 7 and 9 you share only half your voltage with any one inner winding. And the magnetic "coupling" is not totally efficient either to the 3 outside windings. But fundamentally what you suggest does work well enough to figure it all out. In one example on the web, the user supplies 120 VAC (temporarily) to 7-9 and measures ~10V induced across the 1-4 and 3-6 windings. In this manner you can find the 2-5 winding since it's coupled very little to the 7-9 pair (it actually measured out at ~ 1.5V in his video).
Then, like you suggested, he jumpers 6 and 9 to make an autotransformer on that leg and measures the additive 130V across 3-7. Then moves the jumper to 3-9 and measures the subtractive 110V across 6-7.
As you suggested - I'd be more comfortable first tying this with a 12 or 24 volt doorbell transformer, rather than 120 VAC, as long as the doorbell xfmr had enough va to create measurable readings (it might not). His 120 VAC test I assume would start heating up the stator if accidentally left on too long?

People do essentially the same thing with a 6V battery but I guess there it helps to have a "galvanometer" or an old analog meter to read the direction of needle deflection. As you hit 7-9 with the battery and create a magnetic field, you can tell the polarity of the 1-4 coil by watching the direction of the deflection. No need to jumper the windings to create the autotransformer.
Incidentally, the screenshot above is from this video - it's 30 minutes long but is nicely detailed. His on screen interactive drawing skills are pretty good!
Ethan - glad it worked out for you!