Swapped the clone in place of the original. As expected, nothing doing. Tried both drives in the other computer: nothing. Tried the hard drive from the backup machine in the primary machine: that booted. Okay, that tends to narrow it down to only a data problem on the hard drives themselves, rather than something mechanical.
Put the drives back in their normal places. Connected one of the balky drives to the good machine by USB cable: Could read all of the files. That's good news, because it means I probably won't lose much data.
Put one of the balky drives into the primary computer and booted from WinXP system disk. Tried FIXBOOT and FIXMBR with no success. Even did a repair installation of WinXP: still no boot.
I connected one of the balky drives to the good machine again and ran CHKDSK and a virus scan: no errors or viruses. It just won't boot.
Admitting failure, I went to the next-to-last resort, which was the clone the backup computer into one of the balky drives. That worked: the clone booted in the primary machine, although there were a few restarts while it got used to the minor hardware differences between the two Inspiron 600m laptops. (They have different wireless cards, for instance.)
So I guess I won't have to to the last resort of a fresh-from-scratch WinXP installation, and I don't think I've lost any of my data files. (I saved everything in the My Documents directory, which is just about everything there is to save.) But there are various configuration files for bits and pieces of software that were installed on the primary but not the backup machine and for which I can't find the equivalent on the remaining copy of the old machine's now-unbootable drive.
Eventually, when I've convinced myself that I have as much of the old configurations restored as it's possible for me to get, I will cut all ties with the past and clone the now-working primary drive into the non-bootable drive. That's a point of no return, of course, because it will wipe everything on that drive; however, I've got to do it eventually, because it's the only drive that can hold a clone of what's now my primary computer's drive.
I fear that I'm going to have to go to double-redundancy here, with two backup drives for each working drive, and clones always going to the older of the the backups. That way, there's a secondary backup if the first one fails this way. On the bright side, most of my data is safe and sound, and all I have is a bunch more tiresome application reconfiguration to deal with. And a lot of stored passwords that I'm going to have to start remembering again for a while.