In this Letter, the study of a periodic structure composed of gold strips arranged in double-period unit cells, in a symmetric and asymmetric environment, is reported. The spectral maps show that the formation of the plasmonic bandgap and the extraordinary optical transmission are subjected to the proportion between the strip widths. Moreover, when the asymmetric environment is considered, high-transmittance and high-absorbance states arise. Hence, by controlling the geometrical parameters of the binary-periodic structure, it is possible to tailor the spectral response of the grating enhancing the desired features and exploiting them for different applications.