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Fullertubes are tubular fullerenes with nanotube-like middle section and fullerene-like endcaps. To understand how this intermediate form between spherical fullerenes and nanotubes is reflected in the vibrational modes, we performed comprehensive studies of IR and Raman spectra of fullertubes C90-D5h, C96-D3d, and C100-D5d. An excellent agreement between experimental and DFT-computed spectra enabled a detailed vibrational assignment and allowed an analysis of the localization degree of the vibrational modes in different parts of fullertubes. Projection analysis was performed to establish an exact numerical correspondence between vibrations of the belt midsection and fullerene headcaps to the modes of nanotubes and fullerene C60-Ih. As a result, we could not only identify fullerene-like and CNT-like vibrations of fullertubes, but also trace their origin in specific vibrational modes of CNT and C60-Ih. IR spectra were found to be dominated by vibrations of fullerene-like caps resembling IR-active modes of C60-Ih, whereas in Raman spectra both caps and belt vibrations are found to be equally active. Unlike the resonance Raman spectra of CNTs, in which only two single-phonon bands are detected, the Raman spectra of fullertubes exhibit several CNT-like vibrations and thus provide additional information on nanotube phonons.