Why Is Paul Atreides a Harkonnen?

Paul Atreides is a Harkonnen through his mother. Lady Jessica is the secret daughter of Baron Vladimir Harkonnen, bred by the Bene Gesserit, so the man who destroyed Paul's father is Paul's own grandfather. Nobody tells Paul. He works it out himself in the desert, the night after his father dies. (Dune)

Jessica is the Baron's daughter#

Jessica was raised by the Bene Gesserit, who keep the parents of their daughters hidden. Her father was Baron Harkonnen. Children of Dune finally spells out the line from the Sisterhood's own breeding records: Jessica out of Tanidia Nerus by the Baron Vladimir Harkonnen. (Children of Dune)

Even at Paul's test with the gom jabbar, at the very start of Dune, the Reverend Mother Gaius Helen Mohiam notices it in his face. She sees his father's black hair and the eyes of the Old Duke, but also the brow line of the maternal grandfather who cannot be named. (Dune)

So Paul carries the blood of both houses in the oldest feud in the Imperium, the Atreides and Harkonnen feud. (Dune)

Why the Bene Gesserit did it#

It was on purpose. For generations the Sisterhood had bred noble bloodlines towards the Kwisatz Haderach, a male Bene Gesserit with powers no woman of the order could reach. Jessica was the Harkonnen half of the plan. (Dune)

She was placed with Duke Leto Atreides as his concubine, with orders to bear him only daughters. One of those daughters was meant to marry a Harkonnen heir, Feyd-Rautha, joining the two enemy houses and, the Sisterhood hoped, producing their Kwisatz Haderach. The page on the breeding programme explains the wider scheme. (Dune)

Jessica broke the plan out of love for Leto. She gave him the son he wanted, and Mohiam is furious about it when they meet. Paul turned out to be the Kwisatz Haderach anyway, a generation early and outside the Sisterhood's control. (Dune)

How Paul finds out#

After the Harkonnens and the Emperor's Sardaukar take Arrakeen and Leto dies, Paul and Jessica escape into the deep desert. That night, in a stilltent, with spice in the air, Paul's prescience opens up. Among the things he suddenly sees is his mother's parentage. Jessica is the Baron's daughter, and he is a Harkonnen by blood. (Dune)

Jessica hadn't known. Hearing it from her son, she has to take in that the man who just destroyed her family is her own father. (Dune)

Why he calls the Baron grandfather#

The same night Paul sees two main branches of his future. In one of them, Herbert writes, he confronted an evil old Baron and said: "Hello, Grandfather." The thought of that path sickens him, and he turns away from it. (Dune)

In the novel that meeting never happens. The Baron dies at the Battle of Arrakeen, killed by Paul's little sister Alia, who stabs him with a poisoned needle and calls him Grandfather as she does it. (Dune)

Denis Villeneuve's Dune: Part Two gives the moment to Paul. After drinking the Water of Life, Paul tells Jessica she is the Baron's daughter. In the final battle he walks into the Emperor's tent, where the Baron is waiting, names him as his grandfather and kills him with a knife. (Dune: Part Two)

What the Harkonnen blood means#

Herbert never makes Paul secretly evil because of his grandfather. But the blood ties the story together. The Baron's nephews are Paul's cousins, and before their duel Paul asks Feyd-Rautha, "Shall we fight, cousin?" The feud ends with a family killing itself. (Dune)

It also shows how little the Bene Gesserit controlled. They wanted a Harkonnen-Atreides child they could steer, and they got one, just not the one they planned. Our reading is that this is Herbert's joke on the Sisterhood: their careful mixing of enemy bloodlines worked perfectly, and that is exactly why they lost control of the result. (Dune)

The rest of the family is laid out in the Atreides family tree. (Dune; Children of Dune)