Further testing revealed very strange behaviour. If I set both devices on the same setting (either both "master" or both "slave") Windows can see the CD-ROM drive but not the LS-120. It also takes a long time to boot up and shut down.
If I set them to diffent settings Windows can see the LS-120 (but not the CD-ROM drive). However it can't access the LS-120 nevertheless.
If I disconnect the CD-ROM drive then the LS-120 works normally.