An integral transform is introduced which allows the construction of boundary controllers and observers for a class of first-order hyperbolic PIDEs with Fredholm integrals. These systems do not have a strict-feedback structure and thus the standard backstepping approach cannot be applied. Sufficient conditions for the existence of the backstepping-forwarding transform are given in terms of spectral properties of some integral operators and, more conservatively but easily verifiable, in terms of the norms of the coefficients in the equations. An explicit transform is given for particular coefficient structures. In the case of strict-feedback systems, the procedure detailed in this paper reduces to the well-known backstepping design. The results are illustrated with numerical simulations.