According to Wikipedia, some members of Tradescantia (e.g., inch plants) are toxic to cats: the sap seems to cause dermatitis (source). However, I can't tell if this applies to all members of the genus. In particular, I recently came into possession of a tradescantia x andersoniana (I think), with variegated purple-white patches on the leaves (source).
I've searched and searched, but I can't find anything about whether this particular variety is toxic to cats.
You are unlikely to find a definitive answer I'm afraid; this area of toxicity in plants is not clearly catalogued. You might want to read this https://plantsarethestrangestpeople.blogspot.com/2009/02/small-rant-about-plant-toxicity-lists.html which kind of makes that clear.
Tradescantia zebrina contains oxalic acid crystals in its stems (though not the leaves), so it's probably best to assume that Tradescantia generally also contain this. But even if your cat ate the stems, they are unlikely to kill the cat, but would cause a painful mouth and stomach upset and probably require a visit to a vet. The important thing about plants which are toxic because of oxalates is that they are painful to eat, so usually any animal or human that tries the plant stops very quickly because of the feeling you're eating broken glass. Equally, since a cat is largely covered in fur, contact dermatitis is less likely, but make sure your plant is situated somewhere your cat is unlikely to lay on it.
In my experience, adult cats only chew indoor plants if they have no access to outdoors; they do need to eat grass (usually longish grass) periodically, which acts as an emetic to clear fur balls. If they can't access grass, they may try houseplants instead, but providing them with cat grass from the pet shop at home should stop this behaviour.
