Satsuki is Natsuki's other self, created by his desire to protect himself. Satsuki first appeared when a young Natsuki was betrayed by a tutor, presumably his violin teacher, by stealing one of his songs to make herself famous. Satsuki only appears when Natsuki takes his glasses off. Putting it back on turns him back.
It is revealed that Satsuki was born from 'Natsuki's despair' in order to protect him from dangerous people that may hurt him, and that Satsuki's purpose was to protect Natsuki from further pain. Satsuki is very protective of Natsuki, going so far as to believe that Natsuki does not need anyone in his life but him, although he changes that belief when he met Haruka.
His introduction is different throughout the manga, games and anime. In the manga, Natsuki is very much aware of Satsuki's existence, even being able to talk to him in his subconscious. In the games, Natsuki was at first ignorant of Satsuki's existence, although he realized who Satsuki was as the game progressed. In the anime, Natsuki is still left in the dark about Satsuki while most people around him already know.
In the anime, he became volatile, rash, and violent, although after everyone's efforts and Haruka's words, Satsuki eventually calmed down and realized that Haruka was genuine in her concern. As such, Satsuki reverted back into Natsuki willingly. Afterwards, Haruka comments that Natsuki's way of writing lyrics seem different and Natsuki says it's because he feels like 'someone has been watching over him his whole life', meaning that although he is still largely unaware of Satsuki, he still feels safe and protected by his efforts.
Natsuki adored Haruka ever since he first saw her and reminded him of his pet, Elizabeth. As the first season progresses, his feelings for her are revealed to be romantic in nature. He believes that she is the only person who understands his music. His other self, Satsuki, was harder to approach for Haruka, as he believed that she only want to be close to Natsuki out of her own gain. He states that she has the same look on her as the person who betrayed Natsuki, thus his distrust of her.
In the end, Haruka's determination to help the both of them, not only Natsuki, has softened Satsuki towards her, calling her interesting and that he could now see why Natsuki likes her so much. In Episode 7 of the second season, while on their way to his commercial shoot, he says he loves her 'so, so much' and wants her to be 'only his Haru-chan.' He is shown later in the episode thanking her, and singing his solo song while thinking of her.
In episode 1 of the third season, he tells her he is looking at something cute, before picking her up, and holding her bridal style. He points out a cute cherry blossom, and looks at it with her. He then tells her that he is happy that they get to see cute things together, and also says that he wished he 'could take her away' where cherry blossoms fall just for them, forever.
In Season 4 Episode 1, after STARISH pulls her away from QUARTET NIGHT, he hold her hand and says that "there is no one else for him(but her)."
He shares a room with Kurusu Syo, whom he claims to be as cute as Haruka and is very affectionate towards him. He enjoys petting him and treating him fondly and does not hesitate when it comes to spending time with him. Syo is also often Natsuki's (unwiling) taste-tester.
Natsuki is also either oblivious to Syo's dislike of how Natsuki treats him or chooses to ignore Syo's protests. However, Syo genuinely cares for Natsuki, being the first person to know about his other self 'Satsuki' and giving effort to understand and hide his existence from others.
In Episode 9 of Season 1, Natsuki says, "Syo-chan, you're a cutie and I love you...," whether he is saying he feels genuine love and affection for Syo or loves him as a friend is open to interpretation, but Syo is seen blushing after he hears this and seemed to be at a loss for words.
Natsuki seems to like Tokiya, and treats him as a dear friend. He was shown to be worried and confused when the secret of being HAYATO came out, but he quickly forgave him.
Natsuki seems to treat Ren as he does all of his friends, with openness and affection.
Natsuki seems to like Masato, and asks him when they first meet if he can call him by a 'cute' nickname.
Natsuki is one of the people Otoya turns to for help the most. Once, he was asked to show him about baking, and then proceeded to terrify both Otoya and Syo with his horrendous methods.
The first time Natsuki meets Cecil, he comments on how 'warm' his voice sounds, amazed. When Cecil kisses Haruka's hand, Natsuki quickly becomes jealous, and charges up with the rest of STARISH to defend her when Cecil tires to actually kiss her. He quickly forgives him though, and proceeds to welcome him into the Master Course.