  A rare combination of strong spin-orbit coupling and electron-electron
correlations makes the iridate Mott insulator Sr$_2$IrO$_4$ a promising host
for novel electronic phases of matter. The resemblance of its crystallographic,
magnetic and electronic structures to La$_2$CuO$_4$, as well as the emergence
upon doping of a pseudogap region and a low temperature $d$-wave gap, has
particularly strengthened analogies to cuprate high-$T_c$ superconductors.
However, unlike the cuprate phase diagram that features a plethora of broken
symmetry phases in a pseudogap region that include charge density wave, stripe,
nematic and possibly intra-unit cell loop-current orders, no broken symmetry
phases proximate to the parent antiferromagnetic Mott insulating phase in
Sr$_2$IrO$_4$ have been observed to date, making the comparison of iridate to
cuprate phenomenology incomplete. Using optical second harmonic generation, we
report evidence of a hidden non-dipolar magnetic order in Sr$_2$IrO$_4$ that
breaks both the spatial inversion and rotational symmetries of the underlying
tetragonal lattice. Four distinct domain types corresponding to discrete
90$^{\circ}$ rotated orientations of a pseudovector order parameter are
identified using nonlinear optical microscopy, which is expected from an
electronic phase that possesses the symmetries of a magneto-electric
loop-current order. The onset temperature of this phase is monotonically
suppressed with bulk hole doping, albeit much more weakly than the N\'eel
temperature, revealing an extended region of the phase diagram with purely
hidden order. Driving this hidden phase to its quantum critical point may be a
path to realizing superconductivity in Sr$_2$IrO$_4$.
