We determine the level-shifts induced by cotunneling in a Coulomb blockaded carbon nanotube quantum dot using leading order quasi-degenerate perturbation theory within a single nanotube quartet. It is demonstrated that otherwise degenerate and equally tunnel-coupled K and K′ states are mixed by cotunneling and therefore split up in energy except at the particle/hole-symmetric midpoints of the Coulomb diamonds. In the presence of an external magnetic field, we show that cotunneling induces a gate-dependent g-factor renormalization, and we outline different scenarios which might be observed experimentally, depending on the values of both intrinsic KK′ splitting and spin-orbit coupling.