Why is it important that partial derivatives commute? I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates translates to the requirement that $\Gamma_{ij}^{k} = \Gamma_{ji}^{k}$; which is the covariant derivative version of saying partial derivatives commute: $\nabla_{\partial_i}(\partial_j)=\nabla_{\partial_j}(\partial_i)$.
This is obviously true in the Euclidian settings, and I understand all the details of the proofs. But why is this such an essential property? Why does this capture our intuitive sense of derivatives?
 A: To me, a Riemannian metric and the Levi-Civita connection associated with the metric represent the intrinsic geometric properties of a submanifold in Euclidean space induced by the inner product and natural flat connection on Euclidean space. Since they are intrinsic, their definitions can be extended from submanifolds of Euclidean space to abstract manifolds.
If you don't assume the connection is torsion-free, then there are an infinite number of connections that are compatible with the metric (instead of exactly one), so the link between the geometric properties of the metric and that of the connection is much weaker.
A: The covariant derivative version of trying to commute  partial derivatives is: $\nabla_{\partial_i}\nabla_{\partial_j}-\nabla_{\partial_j}\nabla_{\partial_i} - 0 =
R(\partial_i,\partial_j)$.
Torsion is measuring something different: It is the covariant derivative of the soldering form
$\sigma\in\Omega^1(M,E)$ which you use to identify the vector bundle $E$ with $TM$, where $E$ is the bundle you are considering your covariant derivative on.
A: The Levi-Civita connection is just a very special one - torsion free. It is interesting that the same geometry may be described by switching to another, non-torsion-free connection. E.g. there is such a version of general relativity which is called teleparallel formulation. While the curvature tensor (based on the new connection) vanishes, all the deviation from flatness has been shifted to the torsion tensor (better: vector-valued 2-form). Einstein exchanged his ideas with Cartan in the 1920 about that.
Torison has also an equivalent in physics as dislocation density (disclocations are defects in crystals). The theory has been developed in the 1950's by Kondo, Bilby and Kröner. See also the book Ricci calculus by J.A. Schouten.
To summarize, it is not that important that the connection is symmetric, it is merely a matter of choice. The metric, for instance, is independent of the connection.
A: Here is another way of obtaining the Christoffel symbols  with the symetry imposed by the torsion free condition
$$ \Gamma^i_{k\ell}=\Gamma^i_{\ell k}. $$
This goes  back to Riemann's Habillitation.
Suppose that $(M,g)$ is a Riemann manifold of dimension $N$, $p\in M$. By fixing an orthonormal frame  of $T_pM$ we can find local coordinates $(x^1,\dotsc, x^N)$ near $p$ such that,  $\newcommand{\pa}{\partial} $
$$ x^i(p)=0, \;\; g=\sum_{i,j} g_{ij}(x) dx^i dx^j, $$
$$g_{ij}(x)= \delta_{ij} +\sum_{i,j}\left(\sum_k\pa_{x^k}g_{ij}(0) x^k\right) dx^i dx^j + O(|x|^2). $$
In other words, in these coordinates,
$$ g_{ij}(x)=\delta_{ij} +O(|x|). $$
Riemann was   asking whether one can find    new coordinates near $p$ such that in these coordinates  the metric $g$ satisfies $g_{ij}=\delta_{ij}$. 
As a first step, we can ask whether we can find a new system of coordinates such that,  in these coordinates  the metric $g$ is  described by
$$ g=\sum_{ij}\hat{g}_{ij} dy^idy^j, $$
where
$$\hat{g}(y)=\delta_{ij}+ O(|y|^2). \tag{1} $$
The new coordinates $(y^j)$  are described in terms of the old coordinates $(x^i)$ by a family of Taylor approximations
$$y^j= x^j + \frac{1}{2}\sum_{ij}\gamma^j_{\ell k} x^\ell x^k + O(|x|^3),\;\; \gamma^j_{\ell k}=\gamma^j_{k\ell}. $$
The constraint  (1) implies
$$ \gamma^j_{\ell k}=\frac{1}{2}\left(\pa_{x^\ell}g_{jk}+\pa_{x^\ell}g_{jk}-\pa_{x^j}g_{\ell k}\right)_{x=0}. $$
We see that, in the  $x$ coordinates
$$ \Gamma^i_{k\ell}(p)=\gamma^i_{k\ell},  $$
because $g^{ij}(p)=\delta^{ij}$. 
It took people several decades after Riemann's work to realize that  the coefficients  $\Gamma^i_{k\ell}$ are related to  parallel transport, and ultimately, to a concept of connection.
Ultimately, to my mind, the best    explanation for  the torsion-free requirement comes from Cartan's moving frame technique.   The clincher is the following technical fact:  given a    connection $\nabla$ on $TM$  and a $1$-form $\alpha\in \Omega^1(M)$ then for any vector fields $X,Y$ on $M$ we have
$$d\alpha(X,Y)= X\alpha(Y)-Y\alpha(X)-\alpha([X,Y]) $$
$$= (\nabla_X\alpha)(Y)-(\nabla_Y\alpha)(X)+\alpha(\nabla_XY-\nabla_YX)-\alpha([X,Y]) $$
$$= (\nabla_X\alpha)(Y)-(\nabla_Y\alpha)(X)+\alpha\bigl(\;T_\nabla(X,Y)\;\bigr). $$
If the torsion is zero, the above   equality  looses a term,  and one obtains rather easily Cartan's structural equations of a Riemann manifold.
