1. HeiKazu/KazuHei. This ship is fairly one of the most popular Heizou ships and I think it deserves it's popularity because it'd genuinely good in my eyes. Both of them having a natural understanding of each other and it makes me think how Kazuha is one of the few people Heizou can allow himself to act like his own self in front of. He doesn't needs to worry about asking for help (then throw the idea and handle everything himself), as he knows he can trust Kazuha and knows that Kazuha will help him. It just tells about their trust and closure. They both have dead friends so they can relate to each other. Dead friend club <—— they're part of it wooh

2. Heikuki/Heishino/Shinzou get along according to canon but most of their meetings happen in jail when Shinobu has to boil out Itto and other members of the Arataki gang. But they also have a few backgrounds interactions considering Shinobu asks Heizou to bring Itto (who's in jail at those moments) lavender melons. Maybe they hang out outside of their responsibilities (they probably do knowing the 4.6 event but I feel like all of their conversations were about Lumine/Aether and Heizou was doing the yapping as usual. They're cute but they don't interest me as much as they used to

3. HeiSara/SaraHei are canon co-workers. Though, them being lovers or having something romantic going in is acceptable considering they both are grown ass adults, I can't see them as lovers and can only see them as siblings. They're the siblings both of them needed in their lifes. Both of them are rather distant when it comes to their families (Both of them are the black sheeps of their families despite their success in the Tenryou commision ) and they understand that. They'll understand how it feels like to feel pressured into carrying some duty that they were forced into. The difference is, is that Heizou managed to escape from that pressure (receiving another pressure in the end which is all the detective work and not being able to ask others for help), while Sara continued on carrying and consuming that pressure (receiving another pressure of being the general of the Shogun's army and simply being a reliable general figure. Asking for help would be absurd.). They're similar while not trying to be similar .

4. Ittozou/Heito are pretty silly to be honest. Both of them can be seen as childish and unserious in other people's eyes but if the situation asks for it then they can drop that persona. Both of them aren't liked by other people because of the way Itto acts and Heizou's perspective job at the Tenryou commision (also he's hard to get along with). They both lost friendships that they held dear to their hearts but there's a difference between them. Itto managed to rebuild the friendship that he lost or at least seemed to get his back friend. Something like this can't be said for Heizou. Unlike itto, he will never get this things bavk , since his friend is already dead. He lost his friendship into the abyss of death, the friendship that was given a second chance was lost.

5. AyaHei/Heiyato don't have voicelines for each other, and in the 5.4 event they just stood next to each other. They both work for the government and serve the raiden Shogun (technically). Both of them may be considered to be weird and strange for the way they act. But I think the thing that makes them similar is that both of them didn't had a childhood or they lost it. Heizou, according to the first sentence in his vision story, didn't had a happy childhood and his father was forcing him into martial arts (the thing Heizou didn't enjoy), while Ayato lost his childhood the moment his parents die, as he had to give in to Kamisato's clan duties and become it's soon head. And now they tend to act childish or are seen as such in others eyes. They were adults stuck in children's bodies.

6. Heiyaka/Ayahei have voicelines for each other but I dont recall them meeting. They both come from a perfect family but one is adapted to such life (ayaka), while the other says it's not the life for them (Heizou). I also remember someone mentioning they both hav interest in puzzles and that's something they can bond over.

7. Heirou/Gorouhei is filled with distrust since Gorou tends to not trust Heizou. Heizou seems to be pretty chill with Gorou, so the distrust is one-sided and it comes from Gorou. I can't blame him for not trusting Heizou, since Heizou is someone who comes from the Tenryou commision, the commission they had a war with (that one battle scene where kokomi arrived). Maybe this relationship is about gaining for an "enemy" that doesn't face any dangerous intentions towards the island that you protected for so long. The island you swore to defend.

8. Heikoko/Kokohei may bond over their love for books. Both of them aren't fond of asking for help as they used to be the one people can rely on all the time. They both are definitely mentally exhausted as their duties require thinking and analyzing. They definitely get tired but try to hide it from other people and they're doing their best.

9. Heizuki/Mizou is a Rarepair of mine and that can unite the both of them is that they're pretty secretive about their pasts and their own problems. The only person who they could and managed to possibly open up was the traveller. They both carry the understanding and experience of loosing people who they once called (and maybe still do) friends. And that's something that strives them to do something that they do (Mizuki being a therapist and helping people with their problems, as maybe that way she can prevent further loses. Heizou being a detective who solves crimes and makes other criminals think before they want to commit their crime and also him solving crimes can prevent further loses). The mizuki part might not be canon but I like to imagine it is

10. Heimiya/Yoizou are the two people I think might get along. Yoimiya in general seems to get along with everyone (unlike Heizou who isn't liked by other people thus doesn't gets along with them) and her character is related to festivals. The day Heizou's friend died in his arms was the festival day and I believe Heizou might not like festivals because of that or, at the very least, avoids them (in 5.4 he didn't attend the festival and only stayed in the back. Which could be explained by his Tenryou commission duties...). Maybe Yoimiya can help him overcome that dislike and avoidance for the festivals.

11. HeiThoma/Thomazou are the two people who don't get along canonically. Not in the way you might think- Thoma is respectful towards Heizou but he does says how Heizou is not the easiest person to get along with and this opinion and take is basically the way other people think of Heizou. But Thoma is respectful towards Heizou and I dont think Heizou minds him.

Honorable mention I dont even understand how I could have forgotten about them that time:

12. Heizou x his friend (I'll refer to him as Taiyo) and I need for people to hear me out. They're canon friends, childhood friends even! Heizou has met his friend on the festival and, despite Taiyo's intention to steal from Heizou (perhaps he came from a poor family and stealing was his way towards survival who knows), they became friends pretty quickly. Taiyo was the colourful vision to Heizou's monotone life. Eventually, Heizou realised that his friend was lying to him and he broke their friendship and then later threw all the stolen things Taiyo gave him, leaving a single green pebble to rot under his bed. A year later, Heizou decided to come to a festival only to find his friend getting stabbed. He tried to stop the blood but nothing was working and Taiyo died in his arms that day, leaving Heizou with a green pebble and an anemo vision. Taiyo's death made Heizou ran away from his house and begin his journey as a detective. They're tragic, doomed by the narrative, yaoi, childhood friends, friends, homoerotic relationship between two stupid teenagers (they were definitely acting like idiots around each other) and of COURSE you tried to save him but couldn't wow.