High-order harmonic generation in polyatomic molecules generally involves multiple channels, each associated with a different orbital. Their unambiguous assignment remains a major challenge for high-harmonic spectroscopy. Here we present a multi-moda
l approach, using unaligned SF$_6$ molecules as an example, where we combine methods from extreme-ultraviolet spectroscopy, above-threshold ionization and attosecond metrology. Channel-resolved above-threshold ionization measurements reveal that strong-field ionization opens four channels. Two of them, identified by monitoring the ellipticity dependence of the high-harmonic emission, are found to dominate the harmonic signal. A switch from the HOMO-3 to the HOMO-1 channel at 26 eV is confirmed by a phase jump in the harmonic emission, and by the differing dynamical responses to molecular vibrations. This study demonstrates a modus operandi for extending high-harmonic spectroscopy to polyatomic molecules, where complex attosecond dynamics are expected.
