We show that, apart from deterministic processes, compound Poisson processes with exponential jumps are the only (shifted) non-decreasing Lévy processes whose inverses are also (shifted) non-decreasing Lévy processes.

Processes with independent increments; Lévy processes (msc 60G51), Sample path properties (msc 60G17)
CWI
CWI. Probability, Networks and Algorithms [PNA]

Ferreira, J.A. (2001). A note on inverses of non-decreasing Lévy processes. CWI. Probability, Networks and Algorithms [PNA]. CWI.