Electron motion in chemical bonds occurs on an attosecond timescale. This ultrafast motion can be driven by strong laser fields. Ultrashort asymmetric laser pulses are known to direct electrons to a certain direction. But do symmetric laser pulses destroy symmetry in breaking chemical bonds? Here we answer this question in the affirmative by employing a two-particle coincidence technique to investigate the ionization and fragmentation of H₂ by a long circularly polarized multicycle femtosecond laser pulse. Angular streaking and the coincidence detection of electrons and ions are employed to recover the phase of the electric field, at the instant of ionization and in the molecular frame, revealing a phase-dependent anisotropy in the angular distribution of H⁺ fragments. Our results show that electron localization and asymmetrical breaking of molecular bonds are ubiquitous, even in symmetric laser pulses. The technique we describe is robust and provides a powerful tool for ultrafast science.