Let a=x+y and b=x-y, then f(a,b)=f(x+y,x-y).
a+b=2x and a-b=2y, so x=(a+b)/2 and y=(a-b)/2 and xy=(a²-b²)/4.
Also, because a and b are only parameters and can be replaced by any symbols, f(x,y)=(x²-y²)/4 so f(y,x)=(y²-x²)/4 and f(x,y)=-f(y,x), their mean=(f(x,y)+f(y,x))/2=0.