We study the spectra of finite-length carbon nanotubes (CNTs) of arbitrary chirality. They divide into two classes, which arise because of different rotational symmetries of the low-energy eigenstates. In one of them (the zigzag class), the spinless spectrum is doubly degenerate and the two states can be assigned to different values of the valley degree of freedom. In the other (armchair class), the valley degeneracy is removed and the eigenstates are combinations of both valley states. Recent experimental observation of the valley mixing in ultraclean CNT quantum dots is consistent with our theory for armchair-class CNTs.