  We investigate an unusual symmetry of Fe-based superconductors (FeSCs) and
find novel superconducting pairing structures. FeSCs have a minimal
translational unit cell composed of two Fe atoms due to the staggered positions
of anions with respect to the Fe plane. We study the physical consequences of
the additional glide symmetry that further reduces the unit cell to have only
one Fe atoms. In the regular momentum space, it not only leads to a particular
orbital parity separated spectral function but also dictates orbital parity
distinct pairing structures. Furthermore, it produces accompanying Cooper pairs
of $(\pi,\pi,0)$ momentum, which have a characteristic \textit{odd} form factor
and break time reversal symmetry. Such novel pairing structures explain the
unusual angular modulations of the superconducting gaps on the hole pockets in
recent ARPES and STS experiments.
