Rereading Kakashi Retsuden in Japanese.
Iruka is just the Academy Principal...
....that Kakashi now shares an onsen/hot spring obsession with (yes, Iruka's official hobby is going to hot springs)
...who Kakashi feels comfortable opening up about his struggles with insomnia in idle chats (雑談 was translated to "conversation' in English... not wrong, but lost the nuance)
....and going to him to discuss Hokage work troubles in private
...and Iruka is comfortable enough with him to laugh off Kakashi's concerns and offer advice ("笑い飛ばす" was translated in English just as "laughed", again, not incorrect but lacks nuance)
...where Kakashi remembers Iruka's every word with perfect recollection while in the middle of a stealth operation
....which Kakashi screwed up anyway because he was too distracted with grinning with amusement at the memory of Iruka's proud face (that particular description of him smiling was left out of the English translation entirely)
...that Kakashi knows before he even set foot in the village that Sarada is staying with Iruka (even though that happened after Kakashi had already left the village and he's been communicating urgent events by hawk...I guess it's something important enough to share with him by someone🤷)
...that Kakashi knows is very skilled at cooking now (side note, love that Iruka taught Sarada how to cook one of Kakashi's favorite foods, salted saury/mackerel pike, so thank you, Sasuke Retsuden)
....that Kakashi knows Iruka is good at all sorts of house work and keeps a neat house despite of Iruka living alone as a bachelor like himself
The thing is, after reading through all the official novel materials I can get my hands on... there is no label on Kakashi's relationship to Iruka in anywhere... (unless you count how Kakashi counted Iruka and Naruto and Gai and Tsunade and everyone else as his comrades in his first book).
Throughout the Hiden/Shinden/Retsuden novels, we see that Obito and Gai are described as Kakashi's 親友/closest friends, Team Minato are his important teammates, Team 7 are the students he is proud of, Tenzou is his trusted junior, the younger generation is the future he'd die to protect....and then there's Iruka, someone only mentioned as principal at the Academy, the only person aside from Rin and Ichiraku folks to have definitively been shown Kakashi's bare face, and more importantly, he is the only one who has been shown Kakashi's face because Kakashi chose to, and not because Kakashi had to for one reason or another.
And it really is one thing if they actually don't have a relationship beyond work colleagues, but it is clear based on all the social cues in the manga, anime, and novels that not only do they have a relationship, but it is a very comfortable and familiar one (I say novels because Iruka and Kakashi's relationship gets revisited in the canon novel for Boruto's genin exam story.)
There is just no official label for them.
They are just those furigana "between the lines" type of guys. (Furigana Rubi, by Kazuhiko Inoue/Kakashi and Toshihiko Seki/Iruka)
dededos
blakekathryn
stormborns
pixelskylines
turnnoffyourmind
annaxmalina
germansingergirl
