We report on the nonlinear optical response of a mono-chiral sample of (6,5) single-wall carbon nanotubes by means of broad-band two-color pump-probe spectroscopy with selective excitation of the S11 excitons. By using a moment analysis of the transient spectra, we show that all the nonlinear features can be accurately accounted for by elementary deformations of the linear absorption spectrum. The photo-generation of S11 excitons induces a broadening and a blue shift of both the S11 and S22 excitonic transitions. In contrast, only the S11 transition shows a reduction of oscillator strength, ruling out population up-conversion. These nonlinear signatures result from many-body effects, including phase-space filling, wave-function renormalization and exciton collisions. This framework is sufficient to interpret the magnitude of the observed nonlinearities and stress the importance of intersubband exciton interactions. Remarkably, we show that these intersubband interactions have the same magnitude as the intraband ones and bring the major contribution to the photo-bleaching of the S22 excitonic transition upon S11 excitation through energy shift and broadening.