There are certain Christians who are well convinced that repentance involves only turning to God and not away from sin. Although that may be the case in some instances, it is not always the case. Take for example Nineveh:
Jon 1:2 Arise, go to Nineveh, that great city, and cry against it; for their wickedness is come up before me.
Jonah is specifically sent to Nineveh due to their “wickedness”. It seems that once a particular city reaches some sort of evilness threshold, God takes action. When God’s prophet finally arrives, he starts proclaiming an imminent destruction:
Jon 3:4 And Jonah began to enter into the city a day’s journey, and he cried, and said, Yet forty days, and Nineveh shall be overthrown.
For some reason the people take this to heart. Their immediate response is to fast and mourn. Once word reaches the King, it is apparent that Jonah’s message implied destruction due to sin. The King responds:
Jon 3:7 And he caused it to be proclaimed and published through Nineveh by the decree of the king and his nobles, saying, Let neither man nor beast, herd nor flock, taste any thing: let them not feed, nor drink water:
Jon 3:8 But let man and beast be covered with sackcloth, and cry mightily unto God: yea, let them turn every one from his evil way, and from the violence that is in their hands.
Jon 3:9 Who can tell if God will turn and repent, and turn away from his fierce anger, that we perish not?
Jon 3:10 And God saw their works, that they turned from their evil way; and God repented of the evil, that he had said that he would do unto them; and he did it not.
It seems the particular problem with Nineveh was not their Godlessness. The problem was their wickedness. The king believes so and the text describes God responding to their turn from wickedness. Either they already believed in God as the one true God (unlikely) or the text doesn’t take issue with the subject.
In short, Nineveh was flagged for destruction based on their sin. Their repentance was a repentance away from sin. God was going to physically destroy them due to this sin.
Applying this to other situations: when Jesus preaches repentance and a coming destruction, it is safe to assume that repentance involves turning from sin.