Dan Strohl of custom car site Hemmings.com wrote a great piece on solvent-welding plastics: basically gluing together broken pieces of plastic with a solvent chosen to work best with that type of plastic.
Step one is identifying the plastic. There’s all sorts of plastics out there, and they all respond to different solvents. Urethane Supply Company has a pretty comprehensive guide to identifying plastics, should the piece you’re repairing not have any identification marks on it. Plastic interior trim is more likely than not to be some sort of polyethylene, such as HDPE. Once you’ve identified the type of plastic, round up some scrap plastic of the same type. Fortunately, I had another broken piece of interior trim from the same car in the same nutmeg color.