  The assignment of an exact space group to the tetragonal CH$_3$NH$_3$PbI$_3$
perovskite structure is experimentally challenging and controversial in the
literature. Average orientation of the methylammonium ion that gives symmetry
to the experimental measurement is not captured in a static density functional
theory calculation, although the quasi-I4cm and quasi-I4/mcm structures are
commonly used in calculations. In this work we have developed a methodology to
quantify the hidden symmetry of these structures using group theory, to enable
use of symmetries in understanding spectroscopy and other properties. We study
the approximate symmetry of vibrational modes, including analysis of degenerate
representations, as well as the dielectric, elastic, electro-optic, Born
effective charge, and Raman tensors and the dynamical matrix. Comparing to each
subgroup of the full tetragonal D$_{4h}$, our results show that the quasi-I4cm
is best described by the expected corresponding point group C$_{4v}$, whereas
the quasi-I4/mcm (despite corresponding to point group D$_{4h}$) is best
described by the lower symmetry C$_{2v}$. Our methodology can be useful
generally for analysis of other soft hybrid materials or any approximately
symmetric material.
