Posting this so that others might Google it, feel like less of a dumbass than I do, and save some head-banging-against-wall time.
Noticed that my nightly tape backups were failing. NT Backup was insisting that the requested tape was not available. Double-checked, and it's the right tape. Hmm.
Maybe the tape's had it -- try again with a new one. NT Backup reports no free media available, either. Uh-oh. There is a known issue wherein the media library gets corrupted, and I've seen it happen before. Follow the prescribed steps to blow off the library, re-insert the new tape, wait for the "can I import that?" notification.
After much spelunking, I finally find a helpful Support entry at the Exabyte (who I believe subsumed Ecrix) site.
The key points to note therein are:
If the tape drive is requesting a cleaning, the Removable Storage Manager will not show a tape loaded in the drive.
Note for VXA users: The VXA tape drive does not use its LEDs to indicate when a cleaning is needed while a tape is loaded in the drive. Make sure the VXA drive is empty when checking for the cleaning LED.
In a nutshell: the drive wanted cleaning. NT Backup didn't know this, and the cleaning LED is oh-so-helpfully disabled when a tape is in the drive. Neat.
Grab the cleaning tape, lather, rinse, but don't repeat -- all is well.