I have just replaced the center post, basket drive, and clutch. The washer won't go into a high speed spin. I suspect that I might have turned over the clutch thrust washer. Would this cause this symptom? The transmission is only a few years old. The motor drive coupling is okay, agitation seems okay, etc. Just the spin cycle isn't working right. Does the water level sensor tell when to go to high speed as well as controlling the fill?

yes, it spins (although that's based on the sound, since the lid switch kills it and the brake is pretty powerful) but it never seems to go very fast. And if I remember correctly, when it was working correctly, it would spin a low speed for a while (30 seconds?) then switch over to high speed (probably to wait until most of the water is drained so it won't spill over the top). I presume that the motor runs in one direction to run the agitator and the reverse direction at one of two speeds to spin. What switches it to the high speed? And I don't hear any transition now. It just seems to run in low speed for the remainder of the cycle.

I don't think the previous reply applies to this washer. This one runs the motor in reverse, slow while doing the initial drain, then goes to high speed. I can hear the transition while the motor briefly stops. And now that it's working again, I can see that by defeating the lid switch.
But as it turns out, my problem was it didn't spin at all, now that I've actually had time to spend with it. I took it apart, took out the now deformed thrust washer and (after reshaping it) turned it over. Reassembled it and it still didn't spin. So, on a whim, I reached in and moved the basket... it continued to turn and gradually built up to full speed. So I now suspect that I had glazed the clutch ring friction pads and it couldn't develop enough friction to release the basket brake and turn the basket. Probably caused by having the clutch and basket assembly locked together incorrectly while the thrust washer was upside down. Didn't even need to take it apart yet again.
Thanks to the person who replied with some helpful information. We now know that installing the thrust washer upside down leads to a no spin symptom. I might note that installing the wrong clutch spring from the set supplied could possibly cause this kind of symptom as well.