H2, the smallest and most abundant molecule in the universe, has a perfectly symmetric ground state. What does it take to break this symmetry? We found that the inversion symmetry can be broken by absorption of a linearly polarized photon, which itself has inversion symmetry. In particular, the emission of a photoelectron with subsequent dissociation of the remaining H2+ fragment shows no symmetry with respect to the ionic H+ and neutral H atomic fragments. This lack of symmetry results from the entanglement between symmetric and antisymmetric H2+ states that is caused by autoionization. The mechanisms behind this symmetry breaking are general for all molecules.