Safety
Rin's relationship with Kazuma.
Although Rin was taken in by Kagura's mother, she often came and stayed at Kazuma's home when she was younger. Kazuma was the one who stayed by her side when she was initially hospitalized by her parents and was present when her mother told her not to bother coming back home again. Kazuma likely felt as much pity for Rin that he did for Kyo and would often watch over her. Although Rin's new home would be with Kagura and her mother, she would often sneak away to Kazuma's and stay there for a while.After seeing Tohru leaving Kazuma's dojo on her way there, she demands to know what business she had there. Kazuma tells her that she came to ask the same question she did: is there a way to break the curse and if so, how? Rin is angry and shocked and tells Kazuma that she won't let Tohru get involved.
Some time passes and Rin decides to spend new year's at Kazuma's. When Tohru and Kyo arrive, Kazuma teases Kyo about how he and Rin used to just stand and glare at each other for several minutes at a time while Haru watched passively. He also mentions to Tohru that Rin didn't want to go to the banquet but didn't seem to know where to go, so he stopped over and invited her to join him. Before midnight, Rin already falls asleep in the corner of the living room.
Once Rin has recovered from being locked away in the cat's room, Kazuma invites her to stay at his place for a while. He figures Rin probably would just come stay anyway, even if other arrangements were made for her. He seems very upset to hear that Rin was locked away in the cat's room; if the despair she experienced was the result of three months, I'm sure he can only imagine what a lifetime would do to Kyo.
Rin remains at Kazuma's until the end of the series. Kazuma seems happy to have the company. After the final banquet, Rin overhears Kazuma talking with Tohru and Kyo about their future plans and she calls them all idiots before walking off. Kyo wonders what she's upset about and Kazuma sympathetically smiles, stating that not everyone is in a position where they can move on.
Analysis
My opinion on their relationship.
Although they rarely seem to interact in the series, Kazuma certainly plays a very important role in Rin's life. He provides her with a place she can feel safe. When she can't go home to her parents and when she feels too uncomfortable with staying at Kagura's home, she seems to come to stay with Kazuma. This is probably because of the sympathy and kindness he showed toward her when her parents abandoned her in the hospital.Before her initial hospitalisation, I think Rin had already been familiar with Kazuma and likely visited his dojo often. Haru was his student at an early age so I imagine Rin likely came over to watch Haru practice. Kazuma appears to be the type who enjoys looking after people and his place seems to be a gathering point for many of the zodiac members. He seems to take in those who have nowhere else to go. I think he is overly kind to Rin because her situation probably reminds him of what Kyo went through before he adopted him.
I love that Kazuma seems to go out of his way to invite Rin to his home for new year's when she realises she has nowhere to go... and I love that he takes her into his home fully so she has somewhere safe to stay. She certainly feels comfortable in his home and at the dojo since she easily falls asleep there and frequently returns. I feel that she's comfortable there because Kazuma accepts her for who she is, temper and all. In the chapter where she confronts him about talking to Tohru about the curse, she had entered without taking her shoes off -- a major faux-pas in Japan -- and he didn't really seemed too upset about it.