We investigate cooperative light scattering by an ordered chain of multilevel emitters, which possess two quasi-resonant transitions with parallel dipole moments. Interference between dipole transitions induced by coupling with the vacuum gives rise to so called cross-damping and cross-shifts, that modify the incoherent and coherent dynamics and can be manifest in the spectroscopic properties of the emitted light. We determine the excitation spectrum when the emitters chain is driven by an external laser in the limit in which the dipolar transitions can be described by harmonic oscillators and the scattering is coherent. We show that the interplay of multilevel interference and superradiance can give rise to measurable effects in chains of alkali-metal atoms.