  The inverse-Compton effect (IC) is a widely recognized cooling mechanism for
both relativistic and thermal electrons in various astrophysical environments,
including the intergalactic medium and X-ray emitting plasmas. Its effect on
thermal electrons is however frequently overlooked in theoretical and numerical
models of colliding-wind binaries (CWB). In this article, we provide a
comprehensive investigation of the impact of IC cooling in CWBs, presenting
general results for when the photon fields of the stars dominate the cooling of
the thermal plasma and when shocks at the stagnation point are expected to be
radiative. Our analysis shows that IC cooling is the primary cooling process
for the shocked-wind layer over a significant portion of the relevant parameter
space, particularly in eccentric systems with large wind-momentum ratios, e.g.,
those containing a Wolf-Rayet and O-type star. Using the binary system WR 140
as a case study, we demonstrate that IC cooling leads to a strongly radiative
shocked wind near periastron, which may otherwise remain adiabatic if only
collisional cooling was considered. Our results are further supported by 2D and
3D simulations of wind-wind collisions. Specifically, 3D magnetohydrodynamic
simulations of WR 140 show a significant decrease in hard-X-ray emission around
periastron, in agreement with observations but in contrast to equivalent
simulations that omit IC cooling. A novel method is proposed for constraining
mass-loss rates of both stars in eccentric binaries where the wind-collision
zone switches from adiabatic to radiative approaching periastron. IC scattering
is an important cooling process in the thermal plasma of CWBs.
