The self-assembly of asymmetric diblock copolymers confined within cylindrical pores is studied using the self-consistent-field theory. The cylinder-forming asymmetric diblock copolymer is chosen to be near the cylinder-gyroid phase boundary in the intermediate segregation region. This choice makes the self-assembled cylindrical structure highly deformable, leading to very rich morphologies under confinement. A rich variety of structures, such as helices, stacked toroids, and perforated tubes, is observed as a function of the degree of confinement characterized by the ratio between the pore diameter D and bulk period L (D/L) as well as pore surface-polymer interactions. The origin of these confinement-induced structures is elucidated. The theoretical results are in good agreement with available experimental observations.