contrary

on the contrary and on the other hand

On the contrary is used to contradict – to say that what has been said or suggested is not true. If we want to give the other side of a question, we use on the other hand, not on the contrary. Compare:

• ‘I suppose the job was boring?’ ‘On the contrary, it was really exciting.’
  The job was boring, but on the other hand it was well paid. (not … on the contrary, it was well paid.)
• He did not make things easy for his parents. On the contrary, he did everything he could to annoy and worry them.
  He did not make things easy for his parents. On the other hand, he could often be wonderfully sweet and loving.

We can emphasise a contrast with in contrast.

• Daytime temperatures in the desert are very high. In contrast, the nights can be bitterly cold. (not On the contrary, the night …)

contrary and opposite

We use opposite (see here), not contrary, to talk about contrasting words.

• ‘Short’ is the opposite of ‘tall’, and also of ‘long’. (not … the contrary of ‘tall’ …)