  Hybrid improper ferroelectricity, where an electrical polarization can be
induced via a trilinear coupling to two non-polar structural distortions of
different symmetry, has recently been experimentally demonstrated for the first
time in the $n$=2 Ruddlesden-Popper compound Ca$_3$Ti$_2$O$_7$. In this paper
we use group theoretic methods and first-principles calculations to identify
possible ferroelectric switching pathways in Ca$_3$Ti$_2$O$_7$. We identify
low-energy paths that reverse the polarization direction by switching via an
orthorhombic twin domain, or via an antipolar structure. We also introduce a
chemically intuitive set of local order parameters to give insight into how
these paths are relevant to switching nucleated at domain walls. Our findings
suggest that switching may proceed via more than one mechanism in this
material.
