  We establish a bijective correspondence between affine connections and a
class of semi-holonomic jets of local diffeomorphisms of the underlying
manifold called symmetry jets in the text. The symmetry jet corresponding to a
torsion free connection consists in the family of $2$-jets of the geodesic
symmetries. Conversely, any connection is described in terms of the geodesic
symmetries by a simple formula involving only the Lie bracket of vector fields.
We then formulate, in terms of the symmetry jet, several aspects of the theory
of affine connections and obtain geometric and intrinsic descriptions of
various related objects involving the gauge groupoid of the frame bundle. In
particular, the property of uniqueness of affine extension admits an equivalent
formulation as the property of existence and uniqueness of a certain groupoid
morphism. Moreover, affine extension may be carried out at all orders and this
allows for a description of the tensors associated to an affine connections,
namely the torsion, the curvature and their covariant derivatives of all
orders, as obstructions for the affine extension to be holonomic. In addition
this framework provides a nice interpretation for the absence of other tensors.
