Layered 5$d$ transition iridium oxides, Sr$_2$(Ir,Rh)O$_4$, are described as unconventional Mott insulators with strong spin-orbit coupling. The undoped compound, Sr$_2$IrO$_4$, is a nearly ideal two-dimensional pseudospin-$1/2$ Heisenberg antiferromagnet, similarly to the insulating parent compound of high-temperature superconducting copper oxides. Using polarized neutron diffraction, we here report a hidden magnetic order in pure and doped Sr$_2$(Ir,Rh)O$_4$, distinct from the usual antiferromagnetic pseudo-spin ordering. We find that time-reversal symmetry is broken while the lattice translation invariance is preserved in the hidden order phase. The onset temperature matches that of the odd-parity hidden order recently highlighted using optical second harmonic generation experiments. The novel magnetic order and broken symmetries can be explained by the loop-current model, previously predicted for the copper oxide superconductors. ; Comment: 7 figures: 3 in main text and 4 in the supplementary materials