I am aware of the following examples of normal surfaces in $\mathbb P^3$ that are projectively isomorphic to their dual varieties:

0. the smooth quadric;

1. Kummer surfaces;

2. The surface with the equation `$x_0^3=x_1x_2x_3$` (in homogeneous coordinates).

What else is known? The base field is algebraically closed of characteristic zero.

Thank you in advance,
Serge