  The $f(T,T_G)$ class of gravitational modification, based on the quadratic
torsion scalar $T$, as well as on the new quartic torsion scalar $T_G$ which is
the teleparallel equivalent of the Gauss-Bonnet term, is a novel theory,
different from both $f(T)$ and $f(R,G)$ ones. We perform a detailed dynamical
analysis of a spatially flat universe governed by the simplest non-trivial
model of $f(T,T_G)$ gravity which does not introduce a new mass scale. We find
that the universe can result in dark-energy dominated, quintessence-like,
cosmological-constant-like or phantom-like solutions, according to the
parameter choices. Additionally, it may result to a dark energy - dark matter
scaling solution, and thus it can alleviate the coincidence problem. Finally,
the analysis "at infinity" reveals that the universe may exhibit future, past,
or intermediate singularities depending on the parameters.
