Let $X$ be a projective variety of dimension $d$ over $k=\bar{k},$ with $L$ an ample line bundle on $X$ and $\eta=c_1(L).$ Hard Lefschetz gives an isomorphism (see BBD) $$ \eta^i:IH^{d-i}(X)\to IH^{d+i}(X) $$ with Tate twist ignored, which, together with the intersection pairing between $IH^{d-i}$ and $IH^{d+i},$ gives a non-degenerate bilinear form $$ IH^n(X)\times IH^n(X)\to(\mathbb Q,\mathbb Q_{\ell},\text{ or }\mathbb C...) $$ for each $n.$

Question: Is it $(-1)^n$-symmetric?

This is so when $X$ is non-singular (which follows from the general fact on "cup products"), or when $n=d.$ The question is related to this MO question Poincaré duality for intersection cohomology. I guess one can probably figured it out by doing some homological algebra on the level of complexes (i.e. before taking hypercohomology groups), and maybe it's written down somewhere.