  We show that in the $f(Q)$ gravity with a non-metricity scalar $Q$, the
curvatures in Einstein's gravity, that is, the Riemann curvature constructed
from the standard Levi-Civita connection, could not be excluded or naturally
appear. The first observation is that even in $f(Q)$ gravity, the conservation
of the matter energy-momentum tensor is not described by the covariant
derivatives in the non-metricity gravity but that is given by the Levi-Civita
connection. The commutator of the covariant derivatives in Einstein's gravity
inevitably induces the Riemann curvature. There is no symmetry nor principle
which prohibits the Riemann curvature in non-metricity gravity. Based on this
observation, we propose and investigate $f\left(Q, \mathcal{G} \right)$ gravity
with the Gauss-Bonnet invariant $\mathcal{G}$ and its generalisations. We show
how $f\left(Q, \mathcal{G} \right)$ models realising any given the
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) spacetime can be reconstructed.
We apply the reconstruction formalism to cosmology. Explicitly, the gravity
models which realise slow roll or constant roll inflation, dark energy epoch as
well as the unification of the inflation and dark energy are found. The
dynamical autonomous system and the gravitational wave in the theory under
investigation are discussed. It is found the condition that the de Sitter
spacetime becomes the (stable) fixed point of the system.
