Many-body entanglement is at the heart of the Kondo effect, which has its hallmark in quantum dots as a zero-bias conductance peak at low temperatures. It signals the emergence of a conducting singlet state formed by a localized dot degree of freedom and conduction electrons. Carbon nanotubes offer the possibility to study the emergence of the Kondo entanglement by tuning many-body correlations with a gate voltage. Here we quantitatively show an undiscovered side of Kondo correlations, which counterintuitively block conduction channels: inelastic cotunneling lines in the magnetospectrum of a carbon nanotube strikingly disappear when tuning the gate voltage. Considering the global \SUT\ $\otimes $ \SUT\ symmetry of a carbon nanotube coupled to leads, we find that only resonances involving flips of the Kramers pseudospins, associated to this symmetry, are observed below the Kondo temperature. Our results demonstrate the robust formation of entangled many-body states with no net pseudospin.