Dan Piponi on Nostr: Part of the story comes from the fact that the derivative operation is anti-self dual ...

Part of the story comes from the fact that the derivative operation is anti-self dual in the sense that the adjoint ∂ᵗ of ∂ is -∂. (This is the true meaning of integration by parts :)

If you're minimising a quadratic function of a vector it's standard to write it as a quadratic form xᵗQx. Eg. if we're minimising (Mx)² we can convert it to standard form as x·MᵗMx. When we write the Lagrangian, which is really a quadratic form, in the usual way we write Lagrangians, we're not writing it in the usual way for quadratic forms.

We when we write T = ½(∂x)·(∂x) we've flipped the sign from the standard form T = -½x·∂ᵗ∂x. This means we write Lagrangians less symmetrically.

This may be more of a mathematical answer and what you want is more physical.