So there are four types of devices in an MPLS network: Provider (P), Provider Edge (PE), Customer Edge (CE) and Customer (C). By basic definition only P and PE devices exchange labeled packets and the connection between PE and CE is characterized by the carrier's labels being fully stripped before being sent to the customer's network.
Carrier serving carrier is usually more analogous to PE's interconnecting, specifically where an additional one (or two) labels are applied to an existing labeled packet to allow it to be carried across the serving carrier's network. When the packet reaches the remote CsC PE these labels are stripped, leaving a labeled packet at egress that is then forwarded across the served carrier's network as normal (including having the remaining labels stripped on a PE-CE connection).