  We investigate the physical effects of translational symmetry breaking in
Fe-based high-temperature superconductors due to alternating anion positions.
In the representative parent compounds, including the newly discovered
Fe-vacancy-ordered $\mathrm{K_{0.8}Fe_{1.6}Se_2}$, an unusual change of orbital
character is found across the one-Fe Brillouin zone upon unfolding the
first-principles band structure and Fermi surfaces, suggesting that covering a
larger one-Fe Brillouin zone is necessary in experiments. Most significantly,
the electron pockets (critical to the magnetism and superconductivity) are
found only created with the broken symmetry, advocating strongly its full
inclusion in future studies, particularly on the debated nodal structures of
the superconducting order parameter.
