Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists?

Start with the observation that any given diffeomorphism $\rho:M\rightarrow M$ of a Riemannian manifold $M$ allows for pulling the metric on $M$ back along the map $\rho$.

Now consider just those Riemannian metrics on a neighborhoods of $0$ in ${\Bbb R}^n$ given by analytic functions $m$ on ${\Bbb R}^n$. $m$ takes values, of course, in real, symmetric, positive-definite $n\times n$ matrices. Assume also (without serious loss of generality) that $m(0)$ equals the identity matrix.

Locally, one can get a new (still analytic) metric by pulling such an $m$ back along any analytic injection $\iota$ from a neighborhood of $0$ into ${\Bbb R}^n$ that fixes $0$. If, furthermore, $\iota$ has its Jacobian at $0$ sitting in the orthogonal group, the new metric at $0$ will still get represented by the identity matrix.

Viewed only up to order $j+1$, these analytic injections $\iota$ will form a finite-dimensional Lie group. This group acts on metrics $m$ viewed up to order $j$ (as one takes derivatives to form the Jacobians needed to pull back metrics).

Now metrics themselves don't form a linear space (taking differences can kill positive definiteness). But (germs of) functions $m - m(0)$ do comprise a linear space, and when viewed only up to order $j$, a finite-dimensional linear space.

So if I said everything right, I have a finite-dimensional Lie group acting on a finite dimensional linear space, and invariant theory should have something to say about classifying the orbits. Metrics in a common orbit will then share all local geometric invariants up to order $k$.