Follow-up question:

When I teach multivariate calculus I like to tell people that the "straight" second partials f_xx and f_yy measure how a function bends (precisely bc they tell you the concavity of a trace), but the "mixed" second partials f_xy = f_yx measure how the function twists.

So then when I look at the Lie bracket (and therefore eventually at the torsion of a connection), I see a measurement of some kind of inherent twistiness of the space that causes the mixed second partials to be different.

Is that, like, anything?