In this paper, we introduce, model and solve the pickup and delivery problem with time windows, multiple stacks, and handling operations (PDPTWMS-H). % In the PDPTWMS-H, a fleet of vehicles based at a depot is used to complete a set of requests which consist of transporting items from a pickup location to a delivery location. The vehicles have multiple compartments operated using last-in-first-out (LIFO) loading which requires the vehicle to be rear-loaded and items can only be unloaded if they are closest to the back door. In the PDPTWMS-H, additional handling operations, referred to as rehandling, are allowed and an additional handling time might be incurred when rehandling items (by unloading and reloading items). The problem consists of determining the number of vehicles and the vehicle routes needed to complete the set of requests at minimal cost while respecting the possible handling operations. We model the PDPTWMS-H with a set-partitioning formulation and resort to branch-price-and-cut (BPC) for its solution. To solve the pricing problem, we derive a labeling algorithm that is able to cope with the different rehandling possibilities. The labeling algorithm keeps track about the information of on-board items such that symmetries with respect to both stacks and item positions are reduced. Extensive tests are performed on benchmark instances to assess the performance of the proposed BPC methodology and to provide insights on the impact of the rehandling flexibility on solution cost and time.