Edit: It seems my question wasn't being read carefully, so I've rewritten it. I've also retagged the question.

1. Let $M$ be a smooth (pseudo-)Riemannian manifold and let $T(M)$ be the smooth tensor bundle on it. Is there a subspace $S \subset T(M)$ closed under (possibly infinitely many) finitary (possibly partially-defined) algebraic operations and finitely generated containing exactly the tensor fields on $M$ which are diffeomorphism-invariant?
2. If the above is an open question because e.g. the diffeomorphism-invariant tensor fields have not been classified, how about asking that $S$ contains only diffeomorphism-invariant tensor fields and in particular contains the metric tensor, inverse metric tensor, the Riemann tensor, the Ricci tensor, the Ricci scalar, and the torsion tensor?
3. Suppose there is such an algebraic structure, and I adjoin an arbitrary tensor field to it. Will there be a non-trivial subgroup of $\mathrm{Diff}(M)$ under which the extended structure is invariant? Conversely, suppose I fix a non-trivial subgroup of $G < \mathrm{Diff}(M)$. Is there an algebraic structure of the same signature invariant under $G$ containing $S$?

Motivation

We can detect, using purely algebraic means in a certain sense, whether an arbitrary complex number $\alpha \in \mathbb{C}$ is an algebraic number: Look for the smallest subfield $\mathbb{Q}(\alpha) \le \mathbb{C}$ containing both $\mathbb{Q}$ and $\alpha$. It is a vector space over $\mathbb{Q}$, and $\alpha$ is algebraic if and only if this vector space is finite-dimensional. Moreover, we can even detect whether $\alpha$ can be expressed in terms of radicals simply by examining the automorphism group of the field extension $\mathbb{Q}(\alpha) / \mathbb{Q}$.

Here, I'm interested in whether or not something similar can be done for tensor fields on a manifold. The first question is analogous the inverse Galois problem — we have a symmetry group, and we are looking for an algebraic structure which is invariant under it. Having found such an algebraic structure, we can ask whether or not there is an analogue of the fundamental theorem of Galois theory, which establishes a bijective correspondence between subextensions of a Galois field extension and the subgroups of its automorphism group.

I'm aware of at least one result in this area, broadly interpreted as an algebraic approach to differential geometry — namely Lovelock's theorem, which classifies all the symmetric divergence-free second-order natural (0, 2)-tensors on a manifold. A corollary of this theorem, I'm told, tells us that the Einstein field equations are essentially unique (in 4 dimensions).