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 in a dark energy-dark matter scaling solution; 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.