Yes! So, when you refer to another sitter with your code (for first time users) you get $10. If someone refers them to you, with their code, they get $10 and you get a new potential client. If you refer someone with your code, and they book you, there isn't really a monetary incentive there, aside from you having booked a client.
As for any-non monetary rewards, I think it helps to have a back up sitter that you can trust for any of your clients when you are not available, as well as for your personal use, should that arise. If you are looking for one, contact a sitter who strikes your interest, tell them you would like them as a back up sitter and actually take the time to get to know them/meet them. When you have a back up sitter you refer someone to, your client is taking your word on that person. So, if things go sour with that person, I do think that it in one way or another may reflect on you (subjective). And if things go great, well, that's fantastic!