Understanding of the fundamental photoresponse of carbon nanotubes has broad implications for various photonic and optoelectronic devices. Here, Z-scan and pump-probe spectroscopy performed across 600-2400 nm were combined to give a broadband 'degenerate' mapping of the nonlinear absorption properties of single-wall carbon nanotubes (SWNTs). In contrast to the views obtained from non-degenerate techniques, sizable saturable absorption is observed from the visible to the near-infrared range, including the spectral regions between semiconducting excitonic peaks and metallic tube transitions. In addition, the broadband mapping unambiguously reveals a photobleaching to photoinduced absorption transition feature within the first semiconducting excitonic band ∼2100 nm, quantitatively marking the long-wavelength cut-off for saturable absorption of the SWNTs investigated. Our findings present a much clearer physical picture of SWNTs' nonlinear absorption characteristics, and help provide updated design guidelines for SWNT based nonlinear optical devices.