SUCCESS!
Many thanks to those who took the time to contribute suggestions. The internet isn't all bad. Just mostly.
Executive Summary: The Reverse Switch was out of adjustment. Adjusting it solved the problem. The PTO (and the rotary cutter attached to it) works as advertised, in forward and reverse. The Reverse Switch is one of the inputs to the A1 Control Module (and who knows how that epoxied black box works?).
Mysteries:
1. Since the rear PTO can operate when the tractor is in reverse, why would the Reverse Switch cause a failure?
2. What the heck is with the 20-30 second delay before it disengages?
Details and kudos:
The Reverse Switch is on the starboard side, very well buried under the seat (which should be removed) and the plastic covers (that should be removed). They are easy to remove -- some push-fasteners and 10mm bolts.
Under the tractor, follow the linkage from the reverse pedal and you will find a plunger-type switch on the starboard side that SHOULD be depressed when the reverse pedal is in the relaxed position. As the reverse pedal is pressed, the plunger is allowed to extend. In this case, when it was not working properly, the plunger was not compressed with the pedal relaxed. Adjusting the switch solved the problem. To do so, however required exposing the switch adjustment (oh, yes and blowing out all of the grass and dirt that had piled up on everything from when it worked normally) and even then, it was tight quarters. There are two bolts that hold the adjustment plate to the . . . transmission? . . . They must be loosened first. Then the switch adjustment bolt needed to be screwed in and thus the switch moved closer to the reverse-pedal linkage until the Reverse Switch is normally depressed, and when the reverse pedal is pressed, the plunger on the switch relaxes and presumably the circuit is open. After that, be sure you tighten the bolts holding the plate to the . . . transmission.
Bottom line - many thanks for those who took the time to help, particularly those who took the time to dive into the system and provide specific guidance. As partial thanks, I hope this post will deliver a quick(er) fix to the next victim. Not counting my time, the fix was FREE.
Long live Right To Repair. We live in the sticks. It's pretty important that we be able to fix our equipment.
Sorry about the photo quality. As I said, it's in tight quarters.
The photos show an adjustment bolt that strikes a plate that moves the reverse switch assembly.
