= Pair symmetry from the normal-coordinate curvature formula
{title2=$R_{ij,kl}=R_{kl,ij}$}
With convention $R(X,Y)=\nabla_{[X,Y]}-[\nabla_X,\nabla_Y]$, and $R_{ij,kl}=g(R(\partial_i,\partial_j)\partial_k,\partial_l)$, normal coordinates give
$$
R_{ij,kl}=\tfrac12(g_{jk,il}+g_{il,jk}-g_{jl,ik}-g_{ik,jl})
$$
at their center. Symmetry of the metric and commutation of second derivatives prove pair exchange. The opposite curvature convention changes the formula's overall sign but preserves all these algebraic symmetries.
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