Charge migration moves electrons from one molecular site to another, in a typical time domain from few hundred attoseconds to few femtoseconds. On this timescale, the nuclei stand practically still, implying that the nuclear point group symmetry is conserved. Because electrons move ultrafast, this can lead to a surprising effect, namely, breaking the spatial symmetry of the electron density in spite of the conservation of nuclear framework symmetry. We demonstrate theoretically that attosecond charge migration achieves this electron symmetry breaking if the electrons are prepared in a coherent superposition of nondegenerate electronic ground and excited states which transform according to different irreducible representations. Two simple examples provide a proofof-principle, namely, periodic attosecond charge migration in the sigma(g) + sigma(u) superposition state of the aligned H-2(+) cation (nuclear point group D-infinity h, but electron symmetry breaking D-infinity h -> C-infinity v) and in the A(1) + B-2 superposition state of the oriented H2O molecule (C-2v vs C-2(v) -> C-s).