  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.
