  We study Banach spaces with the property that, given a finite number of
slices of the unit ball, there exists a direction such that all these slices
contain a line segment of length almost 2 in this direction. This property was
recently named the symmetric strong diameter two property by Abrahamsen,
Nygaard, and P\~oldvere. The symmetric strong diameter two property is not just
formally stronger than the strong diameter two property (finite convex
combinations of slices have diameter 2). We show that the symmetric strong
diameter two property is only preserved by $\ell_\infty$-sums, and working with
weak star slices we show that $\text{Lip}_0(M)$ have the weak star version of
the property for several classes of metric spaces $M$.
