La media vuelta literally means "the half turn," but in this song, it's a phrase loaded with drama and pride.
In this classic bolero, the singer challenges his lover to find someone better. He's so confident she won't that he declares, "Entonces yo daré la media vuelta" (Then I will give the half turn), promising to walk away forever if she does. It’s a powerful metaphor for leaving a situation with dignity and finality.
This song is one of the most confident breakup songs you will ever hear. The singer isn't sad at all. Instead, he tells his lover, "Te vas porque yo quiero que te vayas" [You're leaving because I want you to leave]. He claims total control over the situation, even saying something as bold as "Yo soy tu dueño" [I am your owner]. It's a huge statement of possession and power.
He then challenges his lover to go out and test the waters. He wants her to "conozcas mucha gente" [meet many people] and let "otros labios" [other lips] kiss her. Why? "Para que me compares" [So that you compare me]. He is so certain that no one else can measure up that he's encouraging her to look. He sets an almost impossible condition for his departure: only if she finds someone who loves her more than anyone will he give "la media vuelta" [the half turn] and leave at sunset. It's a dramatic, almost arrogant, declaration of his own worth in the relationship.