  We report magnetic optical second-harmonic generation (SHG) polarimetry and
imaging on Sr$_2$Cu$_3$O$_4$Cl$_2$, which allows direct visualization of the
mesoscopic antiferromagnetic (AFM) structure of a parent cuprate. Temperature-
and magnetic-field-dependent SHG reveals large domains with 90$^{\circ}$
relative orientations that are stabilized by a combination of uniaxial magnetic
anisotropy and the Earth's magnetic field. Below a temperature $T_R$ $\sim$ 97
K, we observe an unusual 90$^{\circ}$ spin reorientation transition, possibly
driven by competing magnetic anisotropies of the two copper sublattices, which
swaps the AFM domain states while preserving the domain structure. This allows
deterministic switching of the AFM states by thermal or laser heating. Near
$T_R$, the domain walls become exceptionally responsive to an applied magnetic
field, with the Earth's field sufficient to completely expel them from the
crystal. Our findings unlock opportunities to study the mesoscopic AFM behavior
of parent cuprates and explore their potential for AFM technologies.
