I am glad you notice that.

I don't need to give mathematical derivation, since this is not a good "explanation".

Intuitively, the hedge ratio we get can be written in to two parts. rho*(sigma(x)/sigma(y)). We see that what we actually hedge is the rho and the volatility ratio. So whatever you do, say use x to hedge y or use y to hedge x, rho is always the same (you do not need to change your position to hedge that since you have already done that), the thing changes is the volatility ratio. This is a reason why it is not a exact inverse. Since only a part of the hedge ratio need to be adjusted.

hope help~

i see your point，thank you