The Einstein field equations are given by (with assuming $\Lambda = 0$), $$ R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}. $$ The principle of general covariance states that the form of these equations are invariant under diffeomorphisms. These equations can be linearized, by introducing $$ g_{ab} = \eta_{ab} + h_{ab} $$ and letting $h_{ab}$ be small. The coordinate change $x'^\mu = x^\mu + \xi^\mu(x^\mu)$ changes the perturbation metric $h$ as $$ h'_{ab} = h_{ab} + \mathcal{L}_\xi\eta_{ab} = h_{ab} + \partial_a \xi_b + \partial_b \xi_a. $$ In the linear theory, which can be readily checked, the Riemann tensor is invariant under the transformation mentioned above, which also means that the energy-momentum tensor $T_{ab}$ is invariant to first order in $h$.
So, now comes my question:
In the full non-linear theory, the gauge freedom is given by general changes of coordinate systems. My understanding goes as follows; the Ricci/Riemann tensors express curvature, which is an intrinsic property of the manifold, and as this curvature is invariant of coordinate changes (or diffeomorphisms?). The "curvature" property of these tensors have to be invariant. Still, they are tensors, and should therefore act as tensors under rotations, boosts, etc., so they can't be invariant either.
I have only seen the gauge transformation properties discussed in the linear theory, and I am having problems understanding what it "means" in this setting to posses an intrinsic property which should be invariant of coordinate system. All of $R_{abcd}$, $R_{ab}$ and $T_{ab}$ posesses some physcial meaningful value, and should therefore have some "invariant part". They should possess some equivalence classes that are different, and not possible to transform into one another by coordinate changes. What does this mean, and what is this "invariant part"? Also, is there a fundamental difference between a gauge transformation/diffeomorphism and a global transformation (such as a rotation/boost)?