  We perform observational confrontation and cosmographic analysis of
$f(T,T_G)$ gravity and cosmology. This higher-order torsional gravity is based
on both the torsion scalar, as well as on the teleparallel equivalent of the
Gauss-Bonnet combination, and gives rise to an effective dark-energy sector
which depends on the extra torsion contributions. We employ observational data
from the Hubble function and Supernova Type Ia Pantheon datasets, applying a
Markov Chain Monte Carlo sampling technique, and we provide the iso-likelihood
contours, as well as the best-fit values for the parameters of the power-law
model. Additionally, we reconstruct the effective dark-energy equation-of-state
parameter, which exhibits a quintessence-like behavior, while in the future the
Universe enters into the phantom regime, before it tends asymptotically to the
cosmological constant value. Furthermore, we perform a detailed cosmographic
analysis, examining the deceleration, jerk, snap and lerk parameters, showing
that the transition to acceleration occurs in the redshift range $ 0.52 \leq
z_{tr} \leq 0.89 $, as well as the preference of the scenario for
quintessence-like behavior. Finally, we apply the Om diagnostic analysis, as a
cross-verification of the obtained behavior.
