Parametrized geodesics determine the symmetric part of an affine connection ($\Delta (v,v)=0⟺\Delta (u,v)+\Delta (v,u)=0$)

Two affine connections have exactly the same geodesics with the same parameters if and only if their difference tensor vanishes on every diagonal pair $(v,v)$. Indeed, their covariant accelerations differ by $\Delta (\dot{\gamma },\dot{\gamma })$; starting a geodesic at each arbitrary initial vector proves necessity. Polarization then says that the symmetric part of the difference is zero. This condition is stronger than agreement of unparametrized geodesic equations.

The difference of two torsion-free connections is symmetric, because subtracting their zero-torsion identities gives $\Delta (X,Y)=\Delta (Y,X)$. Agreement of parametrized geodesics makes this difference skew-symmetric as well, by parametrized geodesics determine the symmetric part of an affine connection. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized geodesics.