  We present two models for the atomic structure of inversion domain boundaries
in wurzite GaN, that have not been discussed in existing literature. Using
density functional theory, we find that one of these models has a lower
formation energy than a previously proposed model known as Holt-$IDB$. Although
this newly proposed model has a formation energy higher that the accepted lower
energy structure, known as $IDB^*$, we argue that it can be formed under
typical growth conditions. We present evidence that it may have been already
observed in experiments, albeit misidentified as Holt-$IDB$. Our analysis was
facilitated by a convenient notation, that we introduced, to characterize these
models; it is based on the mismatch in crystal stacking sequence across the
$\{10\overline{1}0\}$ plane. Additionally, we introduce an improved method to
calculate energies of certain domain walls that challenge the periodic boundary
conditions needed for plane-wave density functional theory methods. This new
method provides improved estimations of domain wall energies.
