A combined experimental and theoretical study of the photophysical properties and excited-state dynamics of semiconducting single-wall carbon nanotubes (SWNTs) is reported. Steady-state and time-resolved fluorescence data as a function of temperature are explained on the basis of a manifold of four low-lying singlet exciton states with kinetically controlled interconversion. Relaxation among these levels is slow and therefore Kasha's rule is not obeyed. Quantum chemical calculations based on time-dependent density functional theory complement the experimental findings. The temperature-dependence of the radiative and nonradiative rate constants are examined.