I happy to report that I rotated the paddle and John Haitt's Bring the Family, which was completely unplayable before, is now playing through without a hitch (Crazy Little Thing Called Love has me dancing around the room). Before I saw your reply, Whitesnake, I assumed you meant the screws that hold the thread in place should be parallel with the antiskate arms, so I twisted the paddle until I could get at the screw on the bottom. After several attempts, I got it rotated to the point where those screws were in fact parallel with the antiskate bars, and I did it without spilling any goo, which was really amazing given my usual ham-fistedness. I was feeling quite proud of myself and looking forward to perfect playback, when I noticed that when I twisted the paddle to get at the screw, I managed to break the thread that goes to the antiskate wheel--much more like my usual work. So it was off to the sporting goods store to buy fishing line. Arrrgh!
I got the old string off, measured out some new line, with a little extra length for insurance, got my footstool over by the silicone cup, put on two pairs of reading glasses for extra magnification, and got ready for frustration and mess. My big worry was that in twisting the paddle to get at the underside of the tiny thread hole, I would break the other thread--I can't see how to get the azimuth knob off, so that would be a real pain in the neck. But I turned that azimuth knob to put extra slack in that thread, which helped. I made a loop in one end of the new fishing line, looped it around the screw on the paddle, and tightened that down. Then I set about fishing the line through the tiny hole in the bottom of the paddle, which I couldn't see--I could only just get the bottom edge of the paddle to clear the edge of the cup. And, of course, it was coated in silicone goo.
Much to my astonishment, I actually got it on the fourth try. I pulled the line up, attached it to the screw on the antiskate bars, readjusted azimuth and tracking force (I had the weight all the way at the end of the arm so the cartridge end would stay safely up in the air), and put on John Haitt with my fingers crossed. I can't begin to tell you the ecstasy I felt hearing it play all the way through without a hitch. And to hear it play with so much weight, dynamics, and tactile immediacy coming out of an absolutely black background was just incredible. (I've read about this black background for years, but this is first time I really heard what they meant.)
I'm not going to officially declare the problem solved until I play a lot more records, but right now I'm cautiously optimistic that it is, indeed fixed. Thank you, thank you, thank you, everyone, for your help with this. Now to start thinking about other cartridges. . .