Relative Langlands duality, as recently introduced by Ben-Zvi--Sakellaridis--Venkatesh, posits that well known formulae in the theory of automorphic periods can be understood as a duality of Hamiltonian actions by Langlands dual groups. In the expository half of the talk, I will introduce relative duality and its relation with fundamental concepts in the Langlands program: automorphic periods, L-functions, and functoriality. In the second half, I will discuss joint work with Akshay Venkatesh in which we follow this philosophy to understand the duality underlying integral representations of L-functions discovered by Garrett and Ginzburg more than 30 years ago. In doing so, we describe a new numerical invariant of L-parameters, generalising Langlands' notion of L-functions.