  The Covariant Canonical Gauge theory of Gravity is generalized by including
at the Lagrangian level all possible quadratic curvature invariants. In this
approach, the covariant Hamiltonian principle and the canonical transformation
framework are applied to derive a Palatini type gauge theory of gravity. The
metric $g_{\mu\nu}$, the affine connection
$\gamma\indices{^{\lambda}_{\mu\nu}}$ and their respective conjugate momenta,
$k^{\mu\nu\sigma}$ and $q\indices{_{\eta}^{\alpha\xi\beta}}$ tensors, are the
independent field components describing the gravity. The metric is the basic
dynamical field, and the connection is the gauge field. The torsion-free and
metricity-compatible version of the space-time Hamiltonian is built from all
possible invariants of the $q\indices{_{\eta}^{\alpha\xi\beta}}$ tensor
components up to second order. These correspond in the Lagrangian picture to
Riemann tensor invariants of the same order. We show that the quadratic tensor
invariant is necessary for constructing the canonical momentum field from the
gauge field derivatives, and hence for transforming between Hamiltonian and
Lagrangian pictures. Moreover, the theory is extended by dropping metric
compatibility and enforcing conformal invariance. This approach could be used
for the quantization of the quadratic curvature theories, as for example in the
case of conformal gravity.
