Short version: Mike's method is exactly the method I use on closed-segment rings.
Long version: This assumes the number of segments in your ring is a multiple of 4.
1. Grab the base in the headstock any way you wish and lock it. Put a chuck with Cole jaws on a live center in the tailstock.
2. Lightly grab your first ring in the Cole jaws using just 4 flathead screws. The edges of the screw heads bite into the ring and hold it very well and the marks will get turned away.
3. Without glue, lock the tailstock so the ring barely touches the base.
4. Turn the ring relative to the base and you will most likely find one place in the rotation where the ring rubs and spins freely 180 degrees later. This is because the ring faces are slightly out of parallel someplace.
5. Rock the ring back and forth within the loose area to find the point where it is still loose, then locate the nearest place your ring to ring alignment provides your desired pattern. Put matching marks on the fixed and moving rings so you can return to that spot easily.
6. Release the tailstock, glue up the ring, and lock the tailstock so that the rings almost touch.
7. Tighten the tailstock quill as you rotate the ring to evenly spread the glue across both faces of the new glue joint.
8. Rotate the ring to get the alignment marks you just made back in alignment then crank the quill down to provide the clamping pressure. This is where the tailstock prevents the ring from sliding eccentric.
9. Release the headstock lock and spin the complete headstock, piece, and tailstock by hand. Be sure your newest ring is running concentric to the headstock. This is why there's a live center in the tailstock.
10. Wipe off the glue squeezeout. Rotating the headstock makes that easy.
11. Optional - After 5 minutes, unlock the tailstock from the bed, back out the Cole jaws to allow you to pull the tailstock back, and wipe off the glue squeezeout on the inside of the new joint.
12. Lock the tailstock back against the newly attached ring and tighten the quill to provide clamping pressure for another few minutes. I find I can't get a new glue joint back apart after a total of 10 minutes of clamping using Titebond Original.
Repeat for each ring. In my shop, there are always tools to put away, floors to sweep, dogs to pet, or coffee to be traded in that few minutes I have to wait while the glue tacks up.