Hi OldSparks.
I believe you are thinking that the 2 hot legs you are bringing to the single phase motor are 120 degrees apart.
Long story short, they are not. I know it's a bit hard to sometimes wrap your head around, but if your circuit has only 2 "wires" (L-L or L-N) they are 180 degrees apart. The 120 degree phase angle you have in your head requires 3 wires, so that you can measure an angle between any 2 of them from the 3rd point.
This is often a source of confusion. Bear with me for a minute for a simple story that might help you visualize.
Imagine you are on a flat 2 dimensional sea. Picture an X-Y graph paper if that helps. There is nothing on any horizon, just endless ocean as far as you can see.
Now imagine that a point pops up on this sea, lets call it L1. There are no other points on this sea, just Line1. So there is nothing yet to measure to or measure from.
Now, suddenly, another point pops up, some distance away. We will call this L2. We measure distance on this sea in volts, and L2 is 208 volts away from L1.
If you draw a straight line to measure the distance between Line1 and Line2, that's all it will be. A straight narrow line. Standing on the point at L1, L2 is 208 volts away, straight down that line (180 degrees). And conversely, somebody standing on L2 would measure L1 also as 208 volts away, but directly the other way, straight down the line. These 2 points will always simply be directly "opposite" each other, straight down the line, the only shape you can draw between 2 points. And that's all the single phase motor cares about... 2 points that have some potential distance (voltage) between them.
When you want to think about 120 degree phase angles in 3 phase systems, it means you want to start drawing triangles (like for a Delta), or maybe a Y shape (for a Wye). But all those shapes require at least 3 points (like needing an L3). If you bring only 2 points onto your big flat sea, you can only draw (and measure) one shape between them... a straight 180 degree line.
I know you want to think that L1 and L2 are 120 degrees apart... and they are... but ONLY if you bring a 3rd point onto that sea to create a shape you can measure angles with. If all you take out of the transformer is 2 wires, then in the motor's little universe it only knows those 2 points.
There is no 3rd point to measure angles from, just the 2 points and the straight (180 degree) line between them.
Apologies in advance if you knew all this already. Hope any of it helps.