I'm a native speaker, but I saw a meme that had it "aquí mando yo" and that sounds weird to me, so I want to know if I'm wrong.
As Charlie notes in his answer, Spanish syntax is rather flexible and the habitual subject-verb-object order can almost always be changed. Sometimes the change has no real meaning; other times it has to do with what Charlie names tematización, which is known in English as topic fronting, i.e. the movement of the topic or theme to the beginning of the sentence. The topic or theme is the thing being talked about and tends to be old information (in which case it can be deleted) or else it appears at the front of the sentence to establish a new context. (That's incidentally why aquí in «Aquí mando yo» is placed at the start of the sentence.)
What happens with yo in «Aquí mando yo» is a complementary phenomenon; the focus or comment (the new information that you introduce about the topic) tends to go at the end of the sentence and also to receive some special stress. The emphasis on this specific sentence is obviously on the subject, yo, so that goes last. An additional effect of placing this focus at the end of the sentence is to make it contrastive: when you say «Aquí mando yo» you are forcefully excluding other hypothetical agents. Indeed, that's why you are using the pronoun, even though the conjugated verb mando would be enough.
Other sentences showing this movement-with-emphasis of the focus could be:
- En la fiesta me saludó tu hermano.
- La organizó Paula.
- Lo que sé me lo contó él.
It is common in Spanish to invert the order of the words in the sentence when answering to direct questions:
—¿Quién está conduciendo el autobús?
—¡No lo está conduciendo nadie!
—¿Quién va a dar la charla sobre programación?
—La va a dar Juan.
—¿Quién manda aquí?
—Aquí mando yo.
As you see, in the question the order is subject-verb-object, but in the answer it is common to reverse it and use an object-verb-subject scheme. This is called sujeto en posición posverbal (or just sujeto posverbal) by the RAE, which states that this is a common thing in Spanish. There are a lot of examples of this usage:
¿Por qué no viene Carlos?
No conviene que te vean.
Cuanto más se enojaba ella, peor se sentía él.
The Spanish language is known to be quite flexible. If the object happens to be more important in the sentence than the subject (which is known as tematización) you can choose to put it first and hence put the subject after the verb:
Muchas de esas historias las han escrito compatriotas vuestros.
There's no absolute right or wrong for choosing between the two patterns. That means that if your opinion is that it is wrong to say, "Aquí mando yo," then you would be wrong!
If "Aquí yo mando" sounds right to your ear, that's okay. It would be interesting for us to know your linguistic background and to what extent it might have been corrupted by prolonged exposure to, say, English.
"Aquí mando yo" is very common in Mexico, at least. But it's okay if it sounds funny to you. Both forms are "correct."
I'll add something about the subtle difference between the two forms. In "Aquí mando yo" the word receiving stress is "yo," and it's in comparison with someone else who might be trying to usurp power. In "Aquí yo mando," I perceive a calmer, but perhaps heavier, statement. It sounds forceful in the way that "You Jane Me Tarzan" does. Certainly effective for getting the point across. (But everything in this paragraph is just intuitive jottings -- nothing scientific -- intended to share some subtle nuances.)