"Joining" points in one mesh Given: a mesh which forms one-n'th of a torus. I Alt-D-duplicate it, rotate the copy by 360°/n, and get the ends to meet up. Repeat n-1 times.

I have now a nicely symmetric object which I can edit further. This is not the problem.

The problem is that when I move a vertex at the border between copies, that nice alignment is destroyed. Obviously I cannot join the corresponding vertices; it's the same underlying mesh, after all..

Thus, how can I keep them in sync while editing? Repeating the edit steps on the "other" side is tedious, and snap-to-vertex doesn't help much either.

To clarify: I want to create a closed object with n-fold symmetry which stays closed and symmetric, no matter which vertex/edge/face I move. A normal torus will not stay symmetric. A 1/nth-torus with n-1 360°/n-rotated copies will not stay closed when I move a vertex/edge/face at the border.
You could merge the vertices by invoking the Remove Doubles operator



In your case you could create the torus without duplicated vertices, starting with an e.g. an 1/8th



By rotating using the 
Spin Tool  Alt-R in Edit Mode