If you’re wanting to go full-blown restoration, follow @SmashTheState ‘s advice. If you just want to get it back on your wall and looking unbroken, it should glue well.
You have all the stars aligned for being able to pull off a good glue joint: the pieces fit together without gaps, the wood is lightweight (therefore porous), and will have broken along grain lines.
CA glues (AKA“super glue”) may not be the best choice. They work best when you can guarantee zero gap in the joint. Wood breaks are rarely that clean. You would certainly need the gel-style glue to keep it from being absorbed into the wood pores before it can cure.
There is a very nice glue, Weldbond , that would do just what you’re hoping for. It dries clear, doesn’t soak into porous wood, ages well, and forms a joint stronger than the surrounding wood.
Your main problem in gluing this piece will be avoiding squeeze-out, excess glue oozing from the joint when you push it together. The Weldbond will help you with this. Here’s the technique: Squeeze some of the glue into a small container (e.g. jar lid). Dilute it with about 5 times as much water. Use a brush (go to Home Depot’s welding section and buy a pack of cheap “flux brushes” for this job; they’re the perfect size and stiffness) to paint this onto all of the surfaces to be joined. Let this dry completely. This is a primer coat that seals the wood grain.
Next, prepare some more glue, but only very slightly diluted to make it possible to brush it on thinly. Brush evenly and thinly onto the two sides of the joint. Allow the glue to dry until it becomes tacky, then fit the joint together and apply hand pressure (no need to clamp). Since the glue is no longer really liquid when the joint is pressed. there isn’t any squeeze-out. If you do happen to get any glue outside the joint, it can be wiped away with a damp rag before it dries.
The joints dry in about an hour, but reach maximum strength in 24 hrs. Because your piece is twisty, you’ll probably have to do one joint at a time, bolstering the piece with rolled towels so the joint can dry without strain.
Good luck!