  In a recent publication a procedure was developed which can be used to derive
completely gauge invariant models from general Lagrangian densities with $N$
order of derivatives and $M$ rank of tensor potential. This procedure was then
used to show that unique models follow for each order, namely classical
electrodynamics for $N = M = 1$ and linearized Gauss-Bonnet gravity for $N = M
= 2$. In this article, the nature of the connection between these two well
explored physical models is further investigated by means of an additional
common property; a complete dual formulation. First we give a review of
Gauss-Bonnet gravity and the dual formulation of classical electrodynamics. The
dual formulation of linearized Gauss-Bonnet gravity is then developed. It is
shown that the dual formulation of linearized Gauss-Bonnet gravity is analogous
to the homogenous half of Maxwell's theory; both have equations of motion
corresponding to the (second) Bianchi identity, built from the dual form of
their respective field strength tensors. In order to have a dually symmetric
counterpart analogous to the non-homogenous half of Maxwell's theory, the first
invariant derived from the procedure in $N = M = 2$ can be introduced. The
complete gauge invariance of a model with respect to Noether's first theorem,
and not just the equation of motion, is a necessary condition for this dual
formulation. We show that this result can be generalized to the higher spin
gauge theories, where the spin-$n$ curvature tensors for all $N = M = n$ are
the field strength tensors for each $n$. These completely gauge invariant
models correspond to the Maxwell-like higher spin gauge theories whose
equations of motion have been well explored in the literature.
