As outlined in this blog post, we can give a geometric interpretation to the Christoffel symbols (of the second kind) as follows: If you take the vector $\partial_i$ and infinitesimally translate this in the direction of $\partial_j$, it will change by $\Gamma_{ij}^k\partial_k$.
For example, consider polar coordinates in the plane $(r,\phi)$:
If we take the vector $\partial_r$ and translate this outwards in the $r$-direction, the vector is unchanged since it still has the same direction and magnitude. Thus $\Gamma_{rr}^r=\Gamma_{rr}^\phi=0$.
If we translate the vector $\partial_r$ in the $\theta$-direction, the change in direction is given by $\frac{1}{r}\partial_{\theta}$, so $\Gamma_{r\theta}^r=0$ and $\Gamma_{r\theta}^\theta=\frac{1}{r}$.
If we calculate these symbols, we find that they are symmetric in the lower two indices. The same goes for the Christoffel symbols for cylindrical and polar coordinates. Why is this? It doesn't seem 'obvious' to me that the change in $\partial_r$ from translation in the $\partial_\theta$ direction should be the same as the change of $\partial_\theta$ in the $\partial_r$ direction. I know that this is a feature of the Levi-Civita connection, and you can see this from its definition in terms of the metric. However, is there a geometric way of looking at this?
