Contrary to expectations:
— He didn't stay home; on the contrary, he went out with his friends.

synonyms: contrarily, contrariwise, to the contrary.

Roget 468: on the contrary, per contra.   

Roget 14: contrarily etc. adj.; contra, contrariwise, per contra, on the contrary, nay rather; vice versa; on the other hand etc. (in compensation) 30.