My headphone cable wound around my clarinet and brought the wrong end crashing to the floor. I broke the tip of a vintage hard rubber mouthpiece that is one of my favorites. I dried my tears, then fixed it in about 1/2 hour. Next day, as proof of my faith (in super-glue), I used it in recording a Sabbath video. It worked fine.
On a great MP, the facing is sacred. Here is a method to restore the facing without altering it - not even by a fraction of a hair. I also use this method to restore old MPs with chips and dents in the facing, and to do re-facing.
Epoxy or CA (super-glue) generally work fine for joining breaks. I used Starbond rubberized CA glue because it bonds incredibly well to hard rubber, but doesn't get brittle-hard. It can be trimmed with a blade or by abrasive. I had two gaps to fill in this piece. They required several applications because CA shrinks as it hardens. If gaps had been larger, I would have used epoxy putty instead (J-B Weld or J-B Kwik). Photos show the damage, the initial bonding, gap filling, and the restoration of the facing.
Excess filler must be finished to conform exactly to the original facing. There must be NO removal of original material. The usual method, using abrasive paper on a flat surface, makes it impossible to see exactly where you are cutting. I found a way to SEE exactly where I'm cutting, by using frosted glass as a transparent abrasive. It's much safer and easier.
I have a piece of translucent frosted (etched) glass that is commonly used for bathroom shelves. It feels like about 800 grit abrasive. It's not as sharp as a true abrasive. It removes material gently, and leaves a smooth finish. By using glass grit-side down and wet, I can see exactly the location and progress of the cut.
I got the idea to use glass to survey the facing surface from a Ridenour YT video. I would use plain glass and fog it with my breath, and see exactly how the facing rolls on the flat surface. I changed to frosted glass because it holds the film of moisture longer. THEN I discovered that it is a perfect abrasive. I can inspect and grind simultaneously!
Photos show the stages of the process - the original break, the initial bond, the filled gaps, then use of the abrasive glass to restore the damaged tip and side rail. It's ... like ... miraculous!