  The underdoped phase diagram of the iron-based superconductors exemplifies
the complexity common to many correlated materials. Indeed, multiple ordered
states that break different symmetries but display comparable transition
temperatures are present. Here, we argue that such a complexity can be
understood within a simple unifying framework. This framework, built to respect
the symmetries of the non-symmorphic space group of the FeAs/Se layer, consists
of primary magnetically-ordered states and their vestigial phases that
intertwine spin and orbital degrees of freedom. All vestigial phases have
Ising-like and zero wave-vector order parameters, described in terms of
composite spin order and exotic orbital-order patterns such as spin-orbital
loop-currents, staggered atomic spin-orbit coupling, and emergent Rashba- and
Dresselhaus-type spin-orbit interactions. Moreover, they host unusual
phenomena, such as the electro-nematic effect, by which electric fields acts as
transverse fields to the nematic order parameter, and the ferro-N\'eel effect,
by which a uniform magnetic field induces N\'eel order. We discuss the
experimental implications of our findings to iron-based superconductors and
possible extensions to other correlated compounds with similar space groups.
