Three words whose function is to underline a contrast. Two of them are overlapping - in Danish as well as in English - and unfortunately not the same two:



Til gengæld is a +/- compensation: you win some, you loose some.

Jeg hjalp ham med matematik. Til gengæld hjalp han mig med engelsk. (I heelped him doing the math; in return he help me doing the English homework.)Til gengæld cannot always be translated without rephrasing:

Han fik en dårlig karakter i matematik. Til gengæld fik han 12 i engelsk. (He got a bad grade in math. However, he got an A in English which somewhat balanced it out .)

Derimod is a noncompensating contrast:

Peter fik gode karakterer. Hans søster var derimod dårlig i skolen. (He got good grades. His sister, on the contrary, didn't do well in school.)

Tværtimod says: It is not like this, it is right opposite:

Han var ikke træt. Tværtimod! Han dansede hele aftenen. (He wasn't tired. On the contrary! He was dancing all night)

Til gengæld or derimod is very often a question of choice: If you want to underline the connection between the two cases, you use til gengæld. If you don't see any compensation/retaliation, you use derimod. Tværtimod does not contrast two cases - it builds a contrast into one case.

Han var ikke helt ædru; derimodtil gengældtværtimod, han havde drukket 10 øl
Peter var ædru. Anna var derimodtil gengældtværtimod så fuld, at hun ikke kunne ligge på gulvet uden at holde fast.
Han drak aldrig alkohol, derimodtil gengældtværtimod røg han hash hver dag.