A functional equation related to symmetry of operators

In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\). We require no additional assumptions on the function \(f\). Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein–Uhlenbeck process.