The coaxial stacking of two single-wall carbon nanotubes (SWCNTs) into a double-wall carbon nanotube (DWCNT), forming a so-called one-dimensional van der Waals structure, leads to synergetic effects that dramatically affect the optical and electronic properties of both layers. In this work, we explore these effects in purified DWCNT samples by combining absorption, wavelength-dependent infrared fluorescence–excitation (PLE), and wavelength-dependent resonant Raman scattering (RRS) spectroscopy. Purified DWCNTs are obtained by careful solubilization that strictly avoids ultrasonication or by electronic-type sorting, both followed by a density gradient ultracentrifugation to remove unwanted SWCNTs that could obscure the DWCNT characterization. Chirality-dependent shifts of the radial breathing mode vibrational frequencies and transition energies of the inner and outer DWCNT walls with respect to their SWCNT analogues are determined by advanced two-dimensional fitting of RRS and PLE data of DWCNT and their reference SWCNT samples. This exhaustive data set verifies that fluorescence from the inner DWCNT walls of well-purified samples is severely quenched through efficient energy transfer from the inner to the outer DWCNT walls. Combined analysis of the PLE and RRS results further reveals that this transfer is dependent on the inner and outer wall chirality, and we identify the specific combinations dominant in our DWCNT samples. These obtained results demonstrate the necessity and value of a combined structural characterization approach including PLE and RRS spectroscopy for bulk DWCNT samples.