#kyoumado

Whether canon provides input on them or not.

Through the course of all Madoka Magica related content - and as far as my memory serves me right -, Madoka Kaname and Kyouko Sakura are probably the pair that shares the least amount of time together. Since both of them usually (hyper-)focus on other characters respectively (or get hyper-focused on by those characters in return), they don't interact much. In fact, I would go as far as to say that they only really get to spend time together when ALL the other girls are out of the picture and/or not available.

For example, in the main story timeline, they only really get to talk after Sayaka had turned into a Witch. It's at this point that Kyouko reaches out, asking Madoka to conduct her plan to pull Sayaka out of her Witch form together - and despite the fact that they haven't had a direct conversation before, Kyouko must still have taken enough note to realize how important Sayaka was to Madoka (and vice versa) to confide in her like that. Then there is the scene in the last volume of The Different Story, where Madoka suggests for them to become "secret friends" until they are both "able to face their true feelings" - which refers to Kyouko's difficult relationship with Mami and to Madoka's inferiority complex towards (and feeling left behind by) Sayaka and Mami. And this is basically summarizes their relationship as a whole:

Under normal circumstances, they aren't familiar with each other and struggle with their respective relationships with Mami and/or Sayaka... And yet, there is this peculiar dynamic between these two that allows them to "see through" each other with surprising ease.

Where Kyouko usually upholds the facade of being selfish, rough and ignorant, Madoka recognizes this attitude as "you seem like a person who likes to pick on others, but that's not actually true". And with basically every other character, we can assume that Kyouko would initially have reacted like a cat thrown into a bathtub, lashing out at them for calling her out like that - but not with Madoka. Madoka is actually one of the few characters she allows herself to let her guard down with almost immediately - maybe because Madoka opening up to her about her own insecurities and complexes towards Mami and Sayaka (TDS) makes Kyouko feel like the Senpai for once. Either way, her offering a "tasty bar" to Madoka in the first scene they talk face to face without anyone else around, impressed by Madoka's sudden surge of determination - is a huge deal, as sharing/offering food is one of Kyouko's greatest love languages. And even her way of criticizing Madoka, telling her not to put herself down and to not rush becoming a Magical Girl before she finds something she really wants to fight for, is a lot softer than how she bickers or argues with the others. She also confesses to Madoka how much she used to love stories of Courage and Love succeeding in the end, getting nostalgic, chatty and even a little sappy; telling her that this is why she wants to believe in them being capable of saving Sayaka together (OG). Or how she feels like she can't get back together with Mami after all the damage she has done (TDS). There is a sense of... Hopeful honesty these two share with each other that is unlike any other combination in the Holy Quintet. Whether it's because they're not as close and thus don't fear getting hurt or judged by the other - or because their personality structures may be just what they needed in the exact moment their paths crossed, is for the viewer/reader to decide.

Another factor may be how their family dynamics mirror and contrast one another (thanks to @jamesthedigidestined who pointed this out to me once!). They're both big siblings, they're both idealizing and looking up to one of their parental figures... The main difference is that Kyouko's wish - made for the sake of someone else - caused tragedy for her family, resulting in her hardening up and secretly cursing herself for her choices and idealism. And even if she may deny it if confronted directly, it's quite obvious that those experiences are exactly why she cannot allow sheltered Madoka to sacrifice her loving, peaceful life for a half-baked wish. In no way, shape or form would she want Madoka to make the same mistake she made. Plus, Madoka being open about fearing to be a hindrance isn't just hitting close to home, but is also a much more genuine attitude in comparison to how the other Magical Girls tend(ed) to mask their flaws. That is most likely why Kyouko - while still being direct and to the point - talks less aggressively to her. In a way, she is projecting the Senpai-attitude Mami used to use on her onto Madoka - while giving it her very own spin.

Unfortunately - aside from the interactions they share in the games -, there isn't much else we can work with. Neither Rebellion Story nor Magia Record are providing much about them and the same seems to go for the other spin-off mangas (at least as far as I'm aware of). However, these moments alone are enough to show that there is a foundation between them. Even if you have to blink at times to find them.

Whether I think why and how they’d work.

While their signature colours pink and red do not clash as extremely as other combinations within the Holy Quintet, their personalities are still quite a contrast - and as implied above, in a way, that may be exactly why they could be good for one another: Madoka may need someone who's direct and harsh to encourage her genuinely (while not being as cryptic as Homura and not as "knightly" as Sayaka), whereas Kyouko may need someone gentle but honest she can take care of while simultaneously allowing herself to be soft around (without feeling looked down upon like she tends to do with Mami).

If Madoka knew about Kyouko's backstory (and drawing the parallels between their family set-ups themselves), you can bet on her own mother-hen personality bursting out of her (in ways Mami couldn't be more proud of). Madoka is a family person - she may have trouble believing in her abilities (or even herself) being anything special, but she has a natural affinity for praising and admiring others. And she's not afraid to push her mother out of her bed in the most loving yet strict way possible (in her Madoka-esque way at least). Thus, her wanting Kyouko to have a peaceful, happy family life (while encouraging her not to slack off) should be seen as a given - so she'd try her best to give her that.

Kyouko on the other hand is probably the only character who doesn't put Madoka on a pedestal: Homura is completely enchanted by (and biased towards) her, Sayaka and Mami do have their vulnerable moments around her and are also able to gently scold her, but also have a tendency to play "roles" in her company ("protective best knight-friend" and "perfect Senpai"). Kyouko absolutely doesn't have a filter, speaks her mind and right from the heart - and as tough as her words can be sometimes, she may also help Madoka's soft heart to get more resilient. For her to become more confident, outgoing, daring, to believe in herself - in the same way Kyouko would, as she could most likely sense Madoka's own sense of Courage and Love buried inside of her from the start. Plus, Madoka is capable of proving to her that a happy family life can indeed exist - so Kyouko would want to protect and preserve that as well. They do have similar initial ideals (fighting for love, the good in the world, to protect their families, friends, etc.). They just "represent" and deal with them in contrasting ways to the outside world due to their different life experiences.

Granted, this means they may have conflict potential as well - because a Madoka in a very insecure state (= too passive/selfless) and a Kyouko in a very overbearing state (= too aggressive/selfish) may not be compatible in a less favourable timeline. Kyouko may get fed up with Madoka's more fidgety attitude, whereas Madoka may not be able to deal with Kyouko's mean side. A lot of sulking and angst may ensue. Plus, as mentioned at the very start of the post, they do tend to gravitate more towards the other girls instead of each other - IF they are around.

However - their dynamic in the main timeline as well as in TDS have shown me that Kyouko can soften up + Madoka can toughen up for them to be on equal/compromising grounds. That's why Madoka can accept the tasty bar as a token of Kyouko's trust easily. They can be honest without judging the other for their differing ways. The time and place just have to be right for them to notice each other.

Whether I’d prefer them as platonic or romantic ship.

Honestly speaking, prior to conducting this post - and even after @jamesthedigidestined had rambled to me about it a few times before -, I actually hadn't taken note of them too much. I was familiar with these two major scenes and of their potential somewhere at the back of my head, but I actually wasn't aware of just how well the family angle allows them to have this complementary dynamic. One that is obviously severely overlooked - mainly due to their lack of screentime. And that's a bummer considering how they may potentially have one of the healthiest bonds among all of them.

Canon does imply that they bond over their shared adoration for Sayaka - even if that may only apply when things are not well between them, which is actually a shame, since it doesn't even allow them to function as a friendly trio; as we see in Rebellion Story, when they're on good terms, Sayaka and Kyouko are too intense and Madoka doesn't really fit in anymore (which is a similar scenario with Mami, Sayaka and Madoka in TDS).