Yes, and the only restriction is that the crossing points connect on the torus. But there are only so many such curves described by a short rule system (I don't know how many), and most either don't meet this criterion, or just aren't symmetric in this sense.
The Hilbert curve can divide a torus, but into two different parts. I think there are combinations of Hilbert curves(e.g. four linked curves in a 2x2 array) that do the trick, but it's a bit of a cheat to have repeated curves. I'll look for another fully non-periodic division. This is the one example I've found