"You know, he's the yin to my yang. He completes me."
So the funny part about Dan repeatedly calling Perry the yin to Doof's yang (besides the fact that he can just freestyle some Greek tragedy level bullshit about Doof & Perry's relationship on the fly for Radio Disney or an episode commentary track) is that at face value you might also be like, okay but why is Doof the "yang" (light) and Perry the "yin" (dark)?
I didn't say good vs evil because that's not what yin and yang are about. As a concept it's one of those terms from another language that have variable meanings in English depending on who's using it and why, but its most enduring interpretation - with the disclaimer that I am not a philosopher, just someone who is far too invested in overthinking relationships in a kid's cartoon - is the idea that they are opposing but complementary forces of the energy that the universe operates on.
It's a balance. According to Lao-Tzu, "everything is embedded in yin and embraces yang; through [vital energy] it reaches [harmony]". Two sides of the same coin, sun and moon, my other half - you get the idea. It's not hard to see why Perry and Doof are complementary but opposite; one can't exist without the other: Perry all but gives up evil when Doof has that brief dalliance with Peter; Perry has exactly two POV songs in the entire show and both of them relate to Doof, one of which explicitly states "[his] life, it seems, is empty" without Doof's scheming and he "gave [Perry's] life heroic cause"; and he straight up has no idea what to do with his life and makes the saddest platypus sound ever when Doof bans thwarting as Tri-Governor in LDOS.
On the other hand, Doof is a lot easier to see - partly because he actually talks (though Perry is actually one of the most facially expressive characters in the show - by necessity - though I digress). He's overt about his affections and very clingy (a result of childhood trauma and abuse/neglect by his parents) because Perry's his only real friend. He all but begs Perry to thwart him in that whale episode, he became really upset when Perry seemed to be helping him in the episode with the Dull-and-Boring-inator but was just using him to fix the problem with Phineas and Ferb, and he can't bear to leave Perry behind in Road to Danville, just to list a few sillier examples off the top of my head. Really, the entire B plot could be used as evidence.
(There's also the fun dynamic of Perry being an animal that has human-level sentience, and Doof being a human that was raised by animals - ocelots - during some of his formative years.)
So they're irrevocably intertwined. One of the most important characters in each other's life, according to Dan from SDCC 2015. The LDOS ending song even puts Perry smack dab in the middle of the Doofenshmirtz family with Heinz, Vanessa, and Norm - two other characters deeply wrapped up with Perry as a result of Doof himself. I mean, look at them:
Yeah, this was an excuse to make you look at one of my favorite scenes and witness Perry's hopelessly fond smile, not sorry.
Anyway, back to yin-yang. They were originally ascribed to just "day" and "night", and eventually came to also mean "movement" (because you work during the day), and "rest" (because you sleep at night). In terms of qi, they embody heaven and earth, and in the collection of essays of the Huainanzi, "yang is generated from yin and yin is generated from yang [...] sometimes there is life, sometimes there is death, that brings the myriad things to completion". Technical details of Chinese philosophy aside (which I'm sure exactly nobody is considering when they call someone the yin to their yang - it's really just the "you complete me" part), the words themselves also have various opposing definitions: positive/negative, active/passive, sun/moon, overt/covert, etc etc etc you get the idea.
In terms of their relationship, Doof has the more active role. He's one who makes the crazy inators and plots elaborate traps and monologues everyday; Perry usually just reacts to that accordingly. Doof play-acts at a lofty, larger-than-life "evil", while Perry keeps him grounded ("you are my rock" and "having you around just makes me feel, you know, safer"). Perry isn't necessarily a wholly passive character, but also, like, a lot of his plot is just Shit Happens and Perry Has to Deal With It. Doof is overt about his "evil" schemes and can't stand even white lies. Perry spends his entire life hiding aspects of himself and keeping secrets from his loved ones, and he has a whole song in CATU about being an unrecognized hero. One of these characters is living in the "dark" more than the other...
Doof is overflowing with emotions - most of them negative, but also love: so much so that his daughter feels smothered, because a lot of that love (while borne of good intentions) is misguided, and Doof fails to actually listen to her; it takes a few rocky starts (helped along by Perry himself) before they get to the point where Vanessa considers her dad a "misunderstood genius" and convinces him to give up evil. On the other hand, Perry keeps his cards close to his chest - it's not that he doesn't know how to emote (he clearly adores his boys in ATSD and he apparently is happy to hang out with Doof as a friend outside of his job enough that he has a whole wallet of photos of them just goofing off together), he doesn't because of necessity. He's better than anyone at separating his work life (fighting evil) from his pet life (pretending to be a mindless animal) from his personal life (how much he loves his family, watching his silly dramatic soaps, having a petty streak and sassing Monogram a bit, is genuine friends with Doof).
As a result, he is full of emotions he rarely gets to express - around the Flynn-Fletchers specifically, because Doof does know he's sentient, which is another reason I think they click. Doof gets the single-minded attention he never had; Perry gets the treatment as an equal he never had. They both gain a friendship, and a new sort of family. There's a part of their life imbalanced when the other isn't present.
The yin and the yang. They complete each other.
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