The polarization dynamics of electrons including multiple nonlinear Compton scattering during the interaction of a circularly polarized ultraintense laser pulse with a counterpropagating ultrarelativistic electron beam is investigated. While electron polarization emerges mostly due to spin-flips at photon emissions, there is a nonradiative contribution to the polarization which stems from the one-loop QED radiative corrections to the self-energy, which admits of a simple physical model. We put forward a method to single out the nonradiative contribution to the polarization, employing the reflection regime of the interaction when the radiation reaction is significant. The polarization of electrons that penetrate in the forward direction through a colliding laser is shown to be dominated by the loop effect, while the reflected electrons are mostly polarized by spin-flips at photon emissions. We confirm this effect by quantum Monte Carlo simulations considering the helicity transfer from the laser field to the electrons, taking into account the opposite sign of the polarizations induced by the nonradiative loop effect and radiative spin-flip. Our Monte Carlo simulations show a polarization signal as high as $\ensuremath{\gtrsim}10%$ from the nonradiative effect, amenable for experimental detection with current technology.