It is a well known result that to convert one into the other you need an extra Ito-correction term which basically is a convexity correction.
I understand that you need this extra term in the Ito case. My problem is that I don't have a good intuition why you lose this extra term exactly at the midpoint when you construct the Stratonovich integral. Why not more to the right or more to the left? Or put another way: Why is the situation always that symmetric and thus independent of the integrated function?
One hunch I have is that it has to do with bounded quadratic variation: Because the quadratic function is symmetric the midpoint automatically balances the left and the right hand side's "distortions". Is this idea correct and if yes, how can you show this behaviour in the definition of the Stratonovich (and the Ito) integral?