One regime where we might see departures from general relativity is at the largest accessible scales, with a natural choice in cosmology being the cosmological horizon (or Hubble) scale. We investigate a single-parameter extension to the standard cosmological model with a different strength of gravity above and below this scale—a “cosmic glitch” in gravity. Cosmic microwave background observations favor weaker superhorizon gravity, easing both the Hubble and clustering tensions with other cosmological data. This compounds evidence for an even stronger glitch during Big Bang nucleosynthesis (from helium abundance observations), suggesting that symmetries of general relativity are maximally violated at the Big Bang, but gradually recovered as we approach the present-day cosmological de Sitter scale, associated with the observed dark energy.