  Most ferroelectric crystals have their own set of unique characteristics and
ammonium sulfate (NH$_4$)$_2$SO$_4$ is no exception. We report on two
previously unidentified features in ammonium sulfate: 1) that there are at
least two successive transitions instead of one occurring at the Curie
temperature $T$$_C$ = 223 K according to dielectric constant measurements; and
2) pronounced step-like anomalies are found in the magnetic susceptibility
exactly at $T$$_C$. To explain these results, we take into account that there
exists a previously unidentified linear coupling between the magnetic and
electric dipole moments of the NH$_4$$^+$ tetrahedra due to their rapid
reorientations and distorted geometry, respectively. The magnetic moments are
small, 0.0016 $\mu$$_B$ for every $C$$_3$ reorientation which involve three
protons (H$^+$) undergoing orbital motion. Nevertheless, short-range
correlations exist in the paraelectric phase because the magnetic moments are
restricted to only point along 14 possible orientations due to the symmetry and
periodic nature of the potential wells. At $T$$_C$, $C$$_2$ reorientations
(involving four protons) are no longer energetically feasible so the reduction
in the degrees of freedom to 8 further enhances the effect of the magnetic
interactions. This triggers long-range ordering of the orbital moments in an
antiferromagnetic configuration along the $ab$-plane, which via
Dzyaloshinskii-Moriya interactions, end up canting slightly toward the $c$-axis
direction. Since there exists two types of inequivalent NH$_4$$^+$ groups that
reorient at different frequencies with temperature and do not have the same
degree of distortion, the emerging polar phase is ferrielectric.
