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A look at Rin's relationship with Kagura.
When Kagura's mother found out what was happening to Rin with her parents, she volunteered to take Rin into her own home. Thus, Kagura and Rin have lived together for quite some time. Despite Kagura's efforts to befriend her, Rin doesn't seem remotely interested in forming any kind of bond with her fellow zodiac member. In fact, she tends to spend most of her time in her room because she can't stand to see Kagura and her mother together; it reminds her too much of what she no longer has.
When Rin is discharged from the hospital early on in the series, Kagura attempts to encourage her to do something together. She tells Rin that everyone is going out to the summer villa, including Hatsuharu and most importantly (in her own opinion), Kyo. Rin coldly tells Kagura to stop waiting for Kyo to get around to returning her feelings and says it's too heart-wrenching to see her act like she does. Kagura is taken aback by this and Rin leaves. After she's gone, Kagura's mother asks if they've fought again and tells her not to let Rin go to the villa, as Akito has gone there. Kagura's visibly concerned but Rin's already left.
When it comes time for Rin's graduation, she's locked away in the cat's room and cannot attend so Kagura goes to her school and accepts her diploma on her behalf. When she tells Shigure and Hatori this, they mention that Rin has been hospitalized. Kagura says that she's really worried about her and hopes she hasn't left for good.
Much later, after Rin's recovered from her confinement and is living at Kazuma's, Kagura comes to visit. She tells Rin that she should have gone to Shigure's with all the other zodiac members; Rin tells her that if she knew Kagura was going to come over, she would have gone. XD; Kagura yells at Rin and says if she keeps denying her desire to spend time with Haru and Tohru, that she'll keep interrogating her all through the night. Rin tells her she'd better not and tells her to get out. xD
Kagura finally crosses a line with Rin when she overhears Tohru confessing to her that she has feelings for Kyo. She bursts in and hits Tohru across the face, telling her that she needs to tell these things to Kyo. Rin loses control when she sees Tohru get hurt and starts strangling Kagura, telling her she's no right to hit Tohru or tell her what to do. Kagura is shocked at this sudden outburst but seeing that she's triggered a reaction, she stops and hugs Rin gently and says she's sorry. Rin calms down immediately. They seem to be able to get along a little better after this, although Rin is still dismissive towards Kagura and her feelings for Kyo.
Analysis
My opinion on their friendship.
I think Rin and Kagura have an interesting relationship with one another. I don't think that I would say they are friends (at least not at the beginning of the series); Rin is incredibly candid with what she thinks about Kagura and her feelings for Kyo. She doesn't really hold back at all and doesn't really seem to care how Kagura will feel. I think Rin really does care about her though; she only tells her to move on because she doesn't want to see her stay miserable over her unrequited love.
Kagura's mother asks if they've been fighting "again," implying that they argue often. Given how we see the two girls interact throughout the series, it's easy to imagine that arguments have been frequent during Rin's stay at their home. She does seem to have some understanding about Rin and the pain she's had to endure. When Rin begins freaking out on her over hitting Tohru, Kagura immediately recognizes that she's triggered Rin's issues about physical abuse. She stops and hugs Rin, apologising to her.
I find myself wondering how much Kagura actually cares about Rin. She expresses concern and tries to get Rin involved in activities but she didn't really seem to notice that Rin was missing for three months. It's common for Rin to be missing for a few days at a time so I can understand that Kagura may not really pay much mind to her absence, but three months? She implies that she heard Rin was hospitalized but it strikes me as very strange that she wouldn't try to find out which hospital Rin was in. If she's concerned enough to pick up Rin's diploma from her high school, one would assume she'd at least try to figure out where Rin had gone or tell someone about her absence sooner.
Near the end of the series, they seem to be getting along better. They still argue, but their arguments are a little less serious than they would have been earlier on.