Let $X$ be a projective complex algebraic variety of dimension $d.$ Does it make sense to ask if properties like $$ (x,y)=(-1)^i(y,x) $$ holds, for $x\in IH^i(X,\mathbb Q)$ and $y\in IH^{2d-i}(X)?$ And if it makes sense, is it true?

**Edit:** For the first question, my concern is that, if the self-duality $D_XIC_X\cong IC_X$ of the intersection complex only ensures that $IH^i(X)$ and $IH^{2d-i}(X)$ are dual, without specifying a particular duality, then it makes no sense to ask if the "product" is graded-commutative. Of course, if it does give a particular isomorphism, then it makes sense.

**Edit:** As I learned from Gabber, the answer is yes (of course I'd take the responsibility for misunderstanding his comment if it turns out to be ...), and it follows from the symmetric pairing
$$
IC_X[-d]\otimes IC_X[-d]\to\mathbb Q_{\ell}
$$
normalized so that on the smooth locus, it is the natural identification. I'll be appreciated if anyone can give a reference on this.