The Prisoner's Dilemma

Call a function $f$ mapping the real numbers to themselves a sign-flipping function if for all $x,y\in \mathbb{R}$,$$(x-y)(f(x)+f(y))=(x+y)f(x-y).$$ For example, the identity function $i(x)=x$ is a sign-flipping function because $(x-y)(x+y)=(x+y)(x-y)$ by commutativity of multiplication. Are there any others? Find all sign-flipping functions.