Dino is the only one who seems to really know exactly what kind of person Hibari is. He is not afraid of Hibari's strength and works on more than one occasion to train him in how to better use it. In the Varia Arc, Dino is the one who trained the reluctant Hibari to face his opponent and remarks that Hibari's strength is beyond his enemy as he strikes it down in one hit. Dino acknowledges Hibari as his student and is confident in his skills. He also seems to be the only person other than Reborn who can convince Hibari to sit back and stay out of a fight, which is an incredible feat in and of itself. Dino is also the only character seen to refer to Hibari by his given name, Kyouya.
In the Future Arc, TYL Dino and TYL Hibari seem to be on slightly better terms, with Dino able to easily convince Hibari to act accordingly. Future Dino also seems to follow Hibari when he is being particularly introverted and leaves the group. Hibari doesn't hate Dino like he does most other "herbivores," though he does seem to get irritated by him sometimes. On rare occasions though, Dino is able to get through to him.
In the Curse of the Rainbow Arc, it seems that Hibari has set his sights on Dino - or rather, on taking Dino out in a fight. When it comes to losses or opponents, Hibari is not one to forget the past. While Hibari does not think of himself as Dino's student, the Cavallone leader does. This is apparent when he interrupts a fight between the Varia family and Hibari in order to save him and attempts to teach him the value in using flexibility in battle. (As he knows Hibari is pigheaded and stubborn, particularly in fighting with just brute force.)