Is there a nice way to recast riemannian geometry in terms of principal bundles?

Metric compatibility is about connections on the orthonormal frame bundle. The torsion of a connection is the exterior covariant derivative of the canonical 1-form on the total space of the frame bundle. Kobayashi and Nomizu’s books mostly use principal bundles as their language of choice.
Everything can be phrased in this language but experience shows that it buys you very little to do so.
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The local aspects of Riemannian geometry largely boil down to the Levi-Civita connection --- which is indeed quite special to the Riemannian setting --- but of course the global aspects have a lot to do with the calculus of variations and elliptic PDE theory.

Anyway, your question is probably best answered via the theory of G-structures.  A G-structure on a manifold M is a principal G-subbundle of the general frame bundle FM, where G is a (usually closed) subgroup of GL(n,R).  Specifying an O(n)-structure on M is equivalent to specifying a Riemannian metric on M.  A connection on an O(n)-structure is then equivalent to a metric-compatible connection on TM.

To each G-structure B in FM, there is a G-equivariant function T\_B called the "intrinsic torsion" (an older name is "structure function") whose domain is B and whose codomain is a particular R-vector space equipped with a G-action that I won't explain here.  The function T\_B is zero if and only if B admits a torsion-free G-connection.  In this language, the existence part of the fundamental lemma of Riemannian geometry states that every O(n)-structure has T\_B = 0.
You're looking for the Atiyah algebroid. Connections can be seen as sections of a short exact sequence("SES"), for instance a principle bundle is a SES given by G → P → M and a connection is given by sections σ : M → P. Levi-Civita connections are sections of a similar SES for vector bundles, I can't remember the details of this. You can use extension/reduction of structure group and frame bundles to get a map between short exact sequences, hence you get a map between sections of principle bundles and Levi-Civita connections
If I'm not mistaken Frederic Schuller covers this in his lectures a bit.

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