Compton vs inverse Compton scattering

What you are seeing: two complementary photon-electron scatterings on the same energy axis. Forward Compton ($E$ photon, electron at rest) gives a downshifted photon by $E' = E / (1 + (E / m_e c^2)(1 - \cos\theta))$. Inverse Compton ($E$ photon, relativistic electron at Lorentz factor $\gamma$) gives a typical upshift by $E_\text{typ} = (4/3) \gamma^2 E$ in the Thomson limit, with a maximum energy at $4 \gamma^2 E / (1 + 4 \gamma E / m_e c^2)$ for backscatter.

Inverse Compton is the workhorse of high-energy astrophysics. A $\gamma = 10^4$ electron up-scatters a CMB photon ($E \sim 6 \times 10^{-4}$ eV) into an X-ray photon at $\sim 100$ keV. Klein-Nishina suppression kicks in when $\gamma E \gtrsim m_e c^2$; the readout shows the regime.