We report on the experimental and theoretical study of the resonant eigenmodes of spiral-type terahertz antennas. The analysis is carried out for a varying number of spiral windings. For larger numbers the structure possesses a self-complementary property which allows the application of the Mushiake principle predicting that the impedance of such structures is half the impedance of free space. This permits to observe an equal and frequency independent reflection and transmission coefficient. This property makes the spiral-type terahertz antenna not only a fascinating example of a medium supporting strong resonances in the long wavelength limit but also a medium which can be easily and reasonably homogenized at higher frequencies. This is in stark contrast to most of the existing metamaterials.