If $(M, g)$ is Riemannian manifold, there is unique connection $\nabla$, called Levi-Civita connection, satisfying the following conditions:

1) Compatibility with Riemannian metric, i.e. $\nabla(g)$=0

2) Symmetricity, i.e. $\nabla\_X(Y)-\nabla\_Y(X)=[X, Y]$

While the former seems quite natural (equally, parallel transform of a vector along a curve does not change its length), what geometric intuition is hidden behind the later?

The only idea I have, except for the fact it makes a compatible connection unique, is that it is in some way similar to the equality of partial derivates along commuting fields.