Twins of conic hexagons

Six points on a conic section define 60 different hexagons and therefore 60 Pascal lines. Each Pascal line passes through three of the 45 intersections of connecting lines of the six given points. Instead of searching for collinear triples (Pascal lines) among these 45 points, we identify and classify all six-tuples among the 45 points that lie on a conic section. These six-tuples will be called Pascal twins of the given six points. It turns out that there are also six-tuples that lie on a conic section that have two points in common with the given six points. These six-tuples are called Siamese Pascal twins for evident reasons.