Throughout the early Harry Potter movies and books, the Boy Who Lived is shown accidentally performing dark magic, but what if those events were the work of the horcrux within him? Harry's magic is shown to be volatile, which is partly why the Dursleys are so terrified of the very idea of him being "different", but there's definitely an element of him not being in control. That may be called magical instinct persevering, but there could also be a darker reason behind it.
Harry's first major magical moment comes at the zoo when he unwittingly removes the glass of the snake's enclosure, miraculously without a wand, but there are other examples throughout of his magic seemingly acting of its own free will. Before the events of The Philosopher's Stone, the books confirm Harry once turned a Muggle teacher's wig blue, for instance. And more tellingly, when Harry is being chased by Dudley's gang of bullies, he finds himself up on a roof, without evidence or real possibility of him having apparated. These things are presented as Harry's magical self-preservation or vengeful defense mechanisms in some cases because they fit the idea of his true identity being suppressed.
The telling thing about the moment when Harry mysteriously leaps to a rooftop - and indeed his unwitting use of parseltongue at the zoo - is that both suggest Dark Magic use. The latter is obvious, but the former is confirmed in the later revelation that only the most highly advanced dark wizards are able to fly unaided. And one compelling theory suggests that the magic used is the expression of the piece of Voldemort's soul within him, defending its vessel the same way the horcrux in the locket fights back against Harry and Ron in The Deathly Hallows.
In several of the early cases of Harry doing magic, it is in response to his being - or at least feeling - attacked. Combining the Parseltongue with vanishing the glass at the zoo is a means to punish Dudley, while flying away is in the name of self-preservation. There's even probably a case to suggest that the horcrux might identify the hatred of magic in the Dursleys that makes them ostracize and flat-out abuse Harry and automatically see Dudley as the enemy. Harry's temperament to his extended family would certainly fit with the added poison of that sort of influence.
Going further than the theory, there's also the possibility that Harry's horcrux wanted to be found by its original master. Acts of magic willfully ignoring the rules against underage magic usage led to punishment, but that sort of dark magic usage would presumably alert anyone in Voldemort's remaining circle. Like the One Ring calling out to Sauron, Harry's fate was always tied to Voldemort because of the dark compulsion he had towards him. Ironically, of course, while the horcrux might have protected him throughout the Harry Potter series, its ultimate destruction was also what saved Harry when Voldemort attempted to kill him in the forest.