  New surprises continue to be revealed about La$_2$CuO$_4$, the parent
compound of the original cuprate superconductor. Here we present neutron
scattering evidence that the structural symmetry is lower than commonly
assumed. The static distortion results in anisotropic Cu-O bonds within the
CuO$_2$ planes; such anisotropy is relevant to pinning charge stripes in
hole-doped samples. Associated with the extra structural modulation is a soft
phonon mode. If this phonon were to soften completely, the resulting change in
CuO$_6$ octahedral tilts would lead to weak ferromagnetism. Hence, we suggest
that this mode may be the "chiral" phonon inferred from recent studies of the
thermal Hall effect. We also note the absence of interaction between the
antiferromagnetic spin waves and low-energy optical phonons, in contrast to
what is observed in hole-doped samples.
