LS-DOS is basically the last version of TRS-DOS 6.x. So yes, LS-DOS is backwards compatible specifically with that. Now if you have any disks that say "LDOS" or TRSDOS 1.3 on them that's completely a different kettle of fish.
Confusing history, if you care:
TRSDOS for the original TRS-80 Model I was rewritten by its original author into a third-party DOS called VTOS after the relationship between its author and Radio Shack went sour. Then a company called Logical Systems, which was intimately connected to a company called Lobo Drives, licensed VTOS, renaming it "LDOS", to use on their line of TRS-80 compatible expansion hardware and, ultimately, a semi-TRS-80-clone computer. LDOS became a popular DOS choice for the TRS-80 Models I and III, enough so that when Radio Shack gave in and started selling hard disk drives for those systems they elected to sell LDOS as the OS to use with them instead of trying to fix the kind of half-***ed TRSDOS they'd written in-house for the Model III. (Which was kind of a weird hybrid of the original Model I DOS and the DOS they'd written for the Model II, a completely different computer.) Thus when Radio Shack decided to upgrade the Model III to the Model 4 they decided to contract the work for the new OS it needed out to Logical Systems instead of making another hash of it in-house. So... TRSDOS 6 was born, based heavily on LDOS 5.1 for the previous computers and using the same disk format, but not program compatible with it because of the many hardware differences between Model III and Model 4 mode. There were a couple versions of this new OS issued under the TRSDOS label, but when the end of the road was clear for the TRS-80 line Radio Shack completely handed the ball back to Logical Systems and thus the last and final version came out under the "LS-DOS" name instead.
Anyway, if you're staring at a pile of disks that came with a Model 4, which also emulates a Model III, here's the rundown:
TRSDOS 1.3 = Model III mode, only compatible with itself
LDOS 5.x = Model III mode, program compatible with TRS-DOS 1.3 (mostly), but you have to jump through a hoop to copy programs over.
TRSDOS 6.x/LS-DOS 6.3 = Model 4 mode, not program compatible with either Model III DOS, data disk compatible with LDOS.
(And if you have any disks that say DOSPLUS/MultiDOS/NEWDOS-80... consider yourself lucky, you've stumbled across the hoard of a true TRS-80 connoisseur! But sorting out compatibility for all these is a huge kettle of fish. Suffice to say these are all Model III DOSes except for a couple rare versions of DOSPLUS and MultiDOS, they're all better than TRSDOS 1.3, but they're kind of off the main "evolutionary tree" that runs through LDOS.)
