Massive stars are often found in binary systems and it has been argued that binary products boost the ionizing radiation of stellar populations. Accurate predictions for binary products are needed to understand and quantify their contribution to Cosmic Reionization. We investigate the contribution of stars stripped in binaries since (1) they are, arguably, the best-understood products of binary evolution, (2) we recently produced the first non-LTE radiative transfer calculations for the atmospheres of these stripped stars that predict their ionizing spectra, and (3) they are very promising sources since they boost the ionizing emission of stellar populations at late times. This allows stellar feedback to clear the surroundings such that a higher fraction of their photons can escape and ionize the intergalactic medium. Combining our detailed predictions for the ionizing spectra with a simple cosmic reionization model, we estimate that stripped stars contributed tens of percent of the photons that caused cosmic reionization of hydrogen, depending on the assumed escape fractions. More importantly, stripped stars harden the ionizing emission. We estimate that the spectral index for the ionizing part of the spectrum can increase to -1 compared to <-2 for single stars. At high redshift, stripped stars and massive single stars combined dominate the HeII-ionizing emission, but we expect active galactic nuclei drive cosmic helium reionization. Further observational consequences we expect are (1) high ionization states for the intergalactic gas surrounding stellar systems, such as CIV and SiIV and (2) additional heating of the intergalactic medium of up to a few thousand Kelvin. Quantifying these warrants the inclusion of accurate models for stripped stars and other binary products in full cosmological simulations.