High-resolution optical spectroscopic observations of unidentified emission bands from the unusual biconical nebula known as the Red Rectangle are reported. The peak wavelengths and the widths of prominent bands near 5799, 5853, and 6616 angstroms decrease with increasing offset from the central A0-type star HD 44179 and, in the limit of large distance from the star, are shown to converge toward the known values for some of the narrower diffuse interstellar absorption bands at 5797, 5850, and 6614 angstroms, The same carriers give rise to both Red Rectangle emission and corresponding diffuse interstellar absorption bands, and these particular bands arise from electronic transitions in gas-phase molecules.