Out of all of his prison mates, he is closest to Uno. They seem to have shared a longer history than the others and also show a great deal of trust in each other. When Hajime breaks several of Jyugo's bones to calm him down during his fit of rage and shows no remorse, Uno is shown to clearly be the angriest out of the three and seems to truly care for Jyugo. The two are the most intelligent of the four and are always shown interacting with each other the most, showing a genuine respect and deep friendship, though Uno is not above pointing out some of Jyugo's flaws.
Hajime trusts Jyugo the least out of his prison mates, seeing the other three more as juveniles and not threatening, especially since their reasons for being in prison don't warrant any genuine harm or concern. He keeps a close eye on Jyugo who often annoys him with his escape attempts.
Jyugo's opinion of him is not clear but he seems to enjoy teasing Hajime and aggravating him with his escape attempts. He also states Hajime shouldn't be fired for Jyugo's constant jailbreaks as it is not Hajime's fault and that he should take the blame. Despite this, Jyugo seems to hold a distrust towards him and prison guards in general, possibly since the one who placed the unbreakable shackles on him was a prison guard himself. When Jyugo went into a fit of rage, Hajime did not hesitate to break several of his bones and send him into a coma to stop him.
He does seem to show some concern towards Hajime, as during the Building 5 Arc, he chose to rescue him from the underground prison and was worried when Hajime told him to unlock his cuffs and cell in front of Samon Gokuu.
The two have had no interaction thus far in the series and it is unknown what their relationship is, but Momoko Hyakushiki describes them as having similar eyes and being similar in their abilities as escape artists. It is stated that Rokudo, Inmate 610, is the shame of Nanba Prison and known as The Eternal Fugitive. He's also the only inmate to have successfully escaped from there.