Kids cheat all the time when they correct their own papers. Teachers know this and factor that temptation into this practice. There is nothing wrong with letting kids correct their own papers and see how and why they made certain mistakes. If the self-correcting is done in coordination with explaining how students should have arrived at the correct answer, then it's fine. In other words, using the correcting of tests as an extension of the learning process is a good learning technique. If a teacher merely hands tests back to kids and reads off the correct answers, there is no real learning going on.
Wanting to get better grades, wanting to do as well as or better than his twin, wanting to please you and his teacher, wanting your approval - how's that for a start in terms of why a nine year-old might consider cheating in this tempting situation?
I would turn this unsettling situation into a moral victory celebration for your son. Reassure him of your belief in his intelligence and let him know how pleased you are with his courage and his honor. Ask him if he thinks that he should have someone else correct his papers for him for a while? Ask him also what you can do to help him out with this dilemma. Make sure that his temptation to cheat is not borne out of his not understanding the coursework and that he has been afraid to admit this lack of understanding.
To this day, I can remember that in Miss Peterson's third grade class that we used to pass our test papers to the person in front of us to be corrected. Everyone corrected another student's paper. Jean Willis sat in front of me and would always voluntarily correct my mistakes for me (she liked me) and give me a 100% on my tests. I never asked her to stop. At our respective ages, your son had a higher moral sense than I. Funny how we remember certain moments of our childhood.