Well Mark, it turned out that you were right: it was a problem having to do with grouped objects. I discovered it in another model [also part of The Sentinel project]. I had several satellites linked to a larger planet. When one of the moon hotpoints wouldn't center inside the planet, I dragged it out of the hierarchy in the sequencer. That settled the problem. I put the hotpoint where I wanted it, made the moon circle, then dragged it back as child to the planet. Problem solved. Thanks.

I'm sending along a pic. of The Sentinel. Every night when I finish up working on him, I feel satisfied. Every morning when I look at it again, I feel like I have to start over. At the very least, the face is only a placeholder, and the hands need lots of work. It's been quite a learning curve.