mg.metric geometry – Are there any special relations among the 27 lines on a cubic surface?

If we consider 3 skew lines and the hyperboloid formed by them, any other line on the hyperboloid intersects these three lines. Moreover, the lines orthogonal to the plane containing a point on the hyperboloid and a line among the 3 skew lines belong to a linear line complex.

In case of a cubic surface, there are two reguli of 3 skew lines on them. Apart from that, are there any other special relations between the 27 lines? I was wondering about this question because I am trying to retrieve a cubic surface by just knowing three skew lines (well I know three more because of the complementary reguli) on it. Is it possible?