  All degrees of freedom related to the torsion scalar can be explored by
analysing, the $f(T,T_G)$ gravity formalism where, $T$ is a torsion scalar and
$T_G$ is the teleparallel counterpart of the Gauss-Bonnet topological invariant
term. The well-known Noether symmetry approach is a useful tool for selecting
models that are motivated at a fundamental level and determining the exact
solution to a given Lagrangian, hence we explore Noether symmetry approach in
$f(T,T_G)$ gravity formalism with three different forms of $f(T,T_G)$ and study
how to establish nontrivial Noether vector form for each one of them. We extend
the analysis made in \cite{capozziello2016noether} for the form
$f(T,T_{G})=b_{0}T_{G}^{k}+t_{0}T^{m}$ and discussed the symmetry for this
model with linear teleparallel equivalent of the Gauss-Bonnet term, followed by
the study of two models containing exponential form of the teleparallel
equivalent of the Gauss-Bonnet term. We have shown that all three cases will
allow us to obtain non-trivial Noether vector which will play an important role
to obtain the exact solutions for the cosmological equations.
