Toushirou Hitsugaya
bean-sized Captain of Division 10
In any Gotei 13 division, the relationship between its Captain and Vice Captain is probably what determines the entire team's personality - both in and out of work. Being the only two leaders of Division 10, the bond between Hitsugaya and Rangiku is an extremely special one - not even their respective "family" can compete or compare with it.
Both Hitsugaya and Rangiku are sympathetic people who won't hesitate to give all they've got to help their friends. However, being the clumsy persona he is, Hitsugaya prefers action of help over words, while Rangiku is much more skillful: not only would she participate directly in solving your problem, she also knows the precise words that would bring tears to your eyes. As a result, Rangiku proves to be a great help to Hitsugaya in both professional and personal matters. She would gladly be the spokesperson of Hitsugaya's mind when time calls for it, and in order for that to happen as smoothly as possible, trust and loyalty are key to their relationship - whether it be the case of Hinamori or Gin, it's clear that Hitsugaya and Rangiku would place each other's need before personal feelings.
Did I mention how fun it is to watch Hitsugaya glaring at Rangiku for skipping work or getting drunk, time after time? These two together make up a hardcore Seireitei special, I daresay.
Vs. Hinamori?
Contrary to popular belief (especially those who had a chance to visit Fleur de Glace), I don't understand how can one compare Hitsugaya's relationship with Rangiku and Hinamori as something... rivalry-like. Rangiku is a partner and Hinamori is a family so it wouldn't work to measure which relationship is more important to Hitsugaya or what so ever.
To elaborate, I don't think Hinamori is a great choice of a partner for Hitsugaya. First of all, Hinamori is not a confident person like Rangiku. She needs someone who's good with words, can occasionally assure her of her own accomplishments at work and trust her ability 100% during battles - she'd prefer someone to rely on her even under unreasonable circumstances (eg. When she can't handle everything on her own) just to boost up her own confidence. While I won't say Hitsugaya doesn't have any faith in Hinamori's skills, he's clearly overprotective whenever Hinamori is involved and would do anything to shield her from danger to the best of his ability. In other words, Hitsugaya doesn't exactly place Hinamori at equal grounds as himself and would not tolerate her to be treated with the same harshness as he is.
However, any bias Hitsugaya might have towards Hinamori doesn't seem to exist between him and Rangiku. While Hitsugaya worries about Rangiku's safety during battle all the same, he's clearly more confident in Rangiku's ability to solve her own problems without his help. In other words, Hitsugaya and Rangiku's relationship, no matter how fluffy it appears during an ordinary day, is strictly reason-based. Not to mention, Rangiku was the one who encouraged Hitsugaya to become a Soul Reaper in the first place, which adds another professional (i.e. mentor-like) layer to their relationship instead of sweet, lukewarm feelings between families.
Vs. Gin?
I've always had this strange theory that Rangiku seeks a shadow of Gin in Hitsugaya's company, which explains why she's one of the very few people (or probably the only person) who can get along with them both. However, the lack of backup material prevented me from writing about it in the open - until Kubo provided us with the gold aka. the -100 arc in March 2008, which featured a series of backstories about Gotei 13 members.
I believe a certain part of Rangiku's loyalty to Hitsugaya is built upon the illusion that Hitsugaya represents a "white" version of Gin, whom she wished to have by her side. The fact that only three Soul Reapers in Bleach history were ever given the title of a prodigy speaks a lot about what Rangiku sees in both Hitsugaya and Gin: both men were called the genius of their time and surpassed Rangiku in career with much ease, which I believe Rangiku must have been proud of - either as a family or as a mentor. However, similar talents don't necessarily achieve the same outcome: while Hitsugaya remained in Gotei 13 as its youngest Captain, Gin took up the hand of evil and is currently trying to destroy the world Rangiku lives in. Being completely caught in the middle of two men's ideals, both of whom Rangiku cares deeply about, it's obvious that a large amount of pain must be dealt with once the final battle arrives. The age-old foe of humankind repeats: why must one's tear flow because of another's overambition?