Thanks Tha fios agaibh. Sounds like it'd be preferable to not take it apart.
Another clue about the motor, if it matters: when I shut it down (either unplug it or hit the stop switch on the controller), it clicks for maybe five seconds before it goes silent. I have no clue why.
Minder, your diagram makes a lot of sense based on what I saw in the wiring box. Here's my diagrammed version again, this time with what I think matches up to your diagram added in (apologies for the lousy drawing ability and the bit of a tangle):
That is, I think the brown source line matches up to the line on the bottom of your diagram (the route to the bottom of the running winding and the route to the centrifugal switch) , the black line over it that turns blue, runs through the capacitor, and comes out brown (smaller gauge than the black wire) corresponds to the top path to the starter winding, and the blue line at the bottom (same gauge as the black wire) corresponds to the top path to the running winding. The only difference is that my wiring box also has an additional black wire (diagrammed just below the ground) - it has a little label on it with the number "2" on it (none of the others have labels). I suspect that this might be some sort of control or sense wire?
Anyway, if my assumptions are correct, then the solution to reverse the motor would be to swap #1 and #2, correct? That should make the circuit like this, right?
Would that be an effective reversal? Or would the fact that the centrifugal switch also reverses also be a problem? I don't really know anything about the centrifugal switch.
If that would work, then there's one other issue that I see: I don't know which wire would be #2 and which would be #3. If I would guess wrong then it would end up making the circuit like this, right?
How bad of a situation would that be? If I'm not mistaken, that would just do nothing, right? If so, then I could just guess, and if the motor does nothing, I guessed wrong, and if it runs in reverse, I guessed right - correct?