We study the SU(4) Kondo effect in carbon nanotube quantum dots, where doubly degenerate orbitals form four-electron "shells." The SU(4) Kondo behavior is investigated for one, two, and three electrons in the topmost shell. While the Kondo state of two electrons is quenched by a magnetic field, in the case of an odd number of electrons two types of SU(2) Kondo effect may survive. Namely, the spin SU(2) state is realized in a magnetic field parallel to the nanotube (inducing primarily orbital splitting). Application of the perpendicular field (inducing Zeeman splitting) results in the orbital SU(2) Kondo effect. © 2007 The American Physical Society.