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

the smooth quadric;

Kummer surfaces;

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