Consider classical General Relativity without the torsion field (the affine connection is already assumed to be symmetric from the start). It is well known that this theory is independent of the coordinates used. The physics stay the same when we perform a passive coordinates change. This is known as general covariance, and any theory could be formulated in this way. It is not a fundamental characteristic of GR.
It is also known (but there's lot of confusion out there about this) that GR is also invariant under active coordinates transformations (also known as diffeomorphisms), which could be interpreted as a kind of gauge transformations, and not just as simple changes of local coordinates. This property describes physics: an active local coordinates transformation "pushes" the spacetime point to another place of the same manifold. Invariance of the theory under such gauge transformations is telling that the physics is the same at any place on the manifold. Any spacetime point is equivalent, to describe the physics, and the theory is background independent. I believe that this is a subtle formulation of translation invariance, in GR.
Please, don't confuse the general property above with some symmetry of the metric (i.e. isometry). I'm considering general spacetimes, not particular solutions with some special symmetry (isotropy, homogeneity, etc).
GR is also incorporating local Lorentz invariance (at any spacetime point), which says that the physics is independent of the local frame used by the observer (accelerating, rotating axes, free fall, ...).
Usually, the full Poincaré group (Lorentz + translations) isn't made local in standard GR: only the homogeneous Lorentz part is made local. I'm wondering if the "missing" translation part is actually subtly included through the diffeomorphism invariance.
So the question is this: Is the local Lorentz invariance + diffeomorphism invariance (active coordinates changes) of classical GR equivalent to the full local Poincaré invariance?
I'm expecting that some will say "NO" to the question above, because full Local Poincaré invariance is supposed to bring torsion into GR (I never saw any convincing proof of this). Torsion (i.e. the antisymmetric part of the affine connection) is usually assumed to vanish trivially from the start in GR, but there is no contradiction in letting it enter the classical GR formulation. I don't see why we need to explicitly "add" the local translation invariance to get torsion. It can already be there in classical GR, and I suspect this is because of the diffeomorphism invariance -- interpreted as a formulation of the local translation invariance.
I never saw that interpretation before, so I need opinions on it. Maybe I'm getting it all wrong!
If the answer is really a big "NO", then how do we describe local translation invariance to explicitly bring torsion into the theory? As far as I know, torsion could be added directly to classical GR, without the need to talk explicitly about local translation invariance.