  The motivating problem addressed by this paper is to describe those
non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to
a general plane is a complete intersection of curves in that plane. One large
class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the
{\em unexpected cone property} ${\mathcal C}(d)$, a special case of the
unexpected hypersurfaces which have been the focus of much recent research.
After an analysis of ${\mathcal C}(d)$ for small $d$, we show that a
non-degenerate set of $9$ points has a general projection that is the complete
intersection of two cubics if and only if the points form a $(3,3)$-grid.
However, in an appendix we describe a set of $24$ points that are not a grid
but nevertheless have the projection property. These points arise from the
$F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$
and $12$ points with the same feature.
