  Recent experiments in optimally hole-doped iron arsenides have revealed a
novel magnetically ordered ground state that preserves tetragonal symmetry,
consistent with either a charge-spin density wave (CSDW), which displays a
non-uniform magnetization, or a spin-vortex crystal (SVC), which displays a
non-collinear magnetization. Here we show that, similarly to the partial
melting of the usual stripe antiferromagnet into a nematic phase, either of
these phases can also melt in two stages. As a result, intermediate
paramagnetic phases with vestigial order appears: a checkerboard charge
density-wave for the CSDW ground state, characterized by an Ising-like order
parameter, and a remarkable spin-vorticity density-wave for the SVC ground
state -- a triplet d-density wave characterized by a vector chiral order
parameter. We propose experimentally detectable signatures of these phases,
show that their fluctuations can enhance the superconducting transition
temperature, and discuss their relevance to other correlated materials.
