Progress at last: it turns out that tensors whose components are the same in all coordinate systems are called "isotropic tensors".
This is helpful since MathWorld has an entry which lists them, and I found some lecture notes which (partially) show that the list is exhaustive...
Well, the Levi-Civita symbol is tensor density, which is artificially made into a tensor by adding a square root of det(g) term to it.
Maybe ignoring such artificial examples would help in answering the question?
Hello all, this is my first post on this forum, though I have been perusing it for a while.
I am currently re-reading through Carroll's text on SR and there is a curious comment on p24 that intrigues me. Carroll says that the *only* tensors in SR which are invariant are the Kronecker delta...