Generally, the phrase used in American English is "On the contrary." However, that was not your question was, "Does to the contrary make sense?" The answer to your question is, yes it does make sense. While, On the contrary is the preferred usage, still to the contrary makes sense. On the other hand, even the French au contraire makes sense!
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Both "to the contrary" and "on the contrary" are correct phrases that can be used to introduce contrasting information or opinions. They are generally interchangeable, with "on the contrary" being slightly more common in casual conversation.
The prefix that can be added to the word "sense" to indicate that something doesn't make sense is "in-".
The prefix for contrary is "anti-".
You can use "contrary" to express an opposite or different view from a previous statement. For example, "Contrary to popular belief, drinking coffee can have health benefits."
The rhyming phrase from French is "au contraire, mon frère" (literally, on the contrary, my brother where brothermay be used in the sense of a close friend).
The word "contrary" in French is "contraire".