Self-dual structures, whose electric and magnetic parameters can be interchanged without causing any change to the structure, are shown to be inherently matched to free space. With proper design, in such structures, the normally incident energy is funneled through arbitrarily thin air gaps that support TEM or quasi-TEM modes. A finite-length self-dual waveguide, as suggested herein, is inherently matched at both input and output interfaces, even though the field within the waveguide may be substantially complex and have its own characteristic polarization. This structure looks promising for several applications, as detailed below.