  Analogs of the high-T$_c$ cuprates have been long sought after in transition
metal oxides. Due to the strong spin-orbit coupling (SOC), the $5d$ perovskite
iridates Sr$_2$IrO$_4$ exhibit a low-energy electronic structure remarkably
similar to the cuprates. Whether a superconducting state exists as in the
cuprates requires understanding the correlated spin-orbit entangled electronic
states. Recent experiments discovered hidden order in the parent and electron
doped iridates, some with striking analogies to the cuprates, including Fermi
surface pockets, Fermi arcs, and pseudogap. Here, we study the correlation and
disorder effects in a five-orbital model derived from the band theory. We find
that the experimental observations are consistent with a $d$-wave spin-orbit
density wave order that breaks the symmetry of a joint two-fold spin-orbital
rotation followed by a lattice translation. There is a Berry phase and a
plaquette spin flux due to spin procession as electrons hop between Ir atoms,
akin to the intersite SOC in quantum spin Hall insulators. The associated
staggered circulating $J_\text{eff}=1/2$ spin current can be probed by advanced
techniques of spin-current detection in spintronics. This electronic order can
emerge spontaneously from the intersite Coulomb interactions between the
spatially extended iridium $5d$ orbitals, turning the metallic state into an
electron doped quasi-2D Dirac semimetal with important implications on the
possible superconducting state suggested by recent experiments.
