  Sr$_2$CuO$_2$Cl$_2$ (SCOC) is a model undoped cuprate with $I4/mmm$
crystallographic symmetry, and a simple magnetic space group $C_Amca$ with
associated magnetic point group $mmm1'$. However, recent second harmonic
spectroscopy in the antiferromagnetic phase has challenged this picture,
suggesting instead a magnetic point group $4/mm'm'$ that co-exists with the
antiferromagnetism and breaks the two orthogonal mirror planes containing the
tetragonal $c$-axis. Here, we analyze the symmetry of SCOC in light of the
second harmonic results, and discuss possible ground states that are consistent
with the data.
