I've never had difficulty mogging looms into other looms before, as long as they're of the same weapon type or armor class. Sometimes one that isn't ideal looks better than the one your spec should be using, and if you've got a full set of them and they only cost a gold apiece to transmog, why not? I've transmogged my fire mage's heirloom int staff into the Burnished Warden Staff without any issue, for example.
However, when I try to mog my ret pally's Burnished set into the holy-oriented Pristine Lightforge loom set, it refuses to let me, telling me "You must be able to equip an item to use its appearance."
I can equip those looms. I quit out of the transmog screen and equipped them right there, just to make sure I could. They're not even meant for different classes, just different specs of the exact same class. Why is it suddenly not letting me mog one mail-to-plate armor piece into another? I'm just trying to make a look that won't clash with a Silvermoon tabard.