In English, "thank you" (with the stress on the you) is often used in response to someone who has just said "thank you", to reciprocate gratitude. For example, someone providing a service usually says "thank you" to the customer benefiting from the service. An appreciative customer would say "thank you" to humbly suggest they should be the one "giving thanks" (for the service).
Example - A waiter or maître d' of a restaurant is saying goodbye to a customer
Waiter: Thank you! (for dining with us)
Diner: No, thank you!
I'm wondering what, if any, the Italian equivalent to this would be? Would it be a te? Would something like the following ever be used?
Waiter: Arrivederci, grazie!
Diner: (No,) a te!