Problem finally solved!
For the record: originally I had installed a combination of a Tekram DC-315U SCSI adapter and a Certance Tapestor20 tape drive. A utility program that came with the SCSI card reported that it was installed correctly, and a Certance utility program (SGTAPE21) reported that the combination of the SCSI adapter and the drive were functioning correctly. However, I kept finding a lot a verify errors (confusingly, for the same files), both when running under Linux and Windows.
To solve the problem, I replaced the Tekram SCSI adapter with a cheap Adaptec card. Since then, I've not had any more errors.
I don't know if there was a manufacturing problem with the Tekram card (if so, why did the utility program report it was functioning properly?) or if there's some fundamental inconsistency with the combination of the Tekram DC-315U SCSI adapter with the Certance Tapestor20 drive (if so, why did Certance's utility program SGTAPE21 report that things were functioning correctly, when they weren't?). All I know is that having changed the SCSI card, backups now work correctly.