Twisted cubics and quadruples of points on cubic surfaces

We study relations in the Grothendieck ring of varieties which connect the Hilbert scheme of points on a cubic hypersurface $Y$ with a certain moduli space of twisted cubic curves on $Y$. These relations are generalizations of the "beautiful" $Y$-$F(Y)$ relation by Galkin and Shinder which connects $Y$ with the Hilbert scheme of two points on $Y$ and the Fano variety $F(Y)$ of lines on $Y$. We concentrate mostly on the case of cubic surfaces. The symmetries of $27$ lines on a smooth cubic surface give a lot of restrictions on possible forms of the relations.