Axial refers to being parallel to the centerline of the pipe expansion joint. Compression is the axial deflection which will shorten the bellows length. Often confusion occurs because thermal expansion in the piping will cause the pipe expansion joint to be compressed. The specification for a pipe expansion joint should always state the movements as they affect the pipe expansion joint, and not as they are produced by the system.
Extension is the axial deflection which stretches the pipe expansion joints. Piping which is operating at temperatures lower than ambient, such as in cryogenic systems, will contract causing the pipe expansion joint to stretch or experience extension.