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opposite in nature or character; diametrically or mutually opposed:
contrary to fact; contrary propositions.
opposite in direction or position:
departures in contrary directions.
being the opposite one of two:
I will make the contrary choice.
unfavorable or adverse.
perverse; stubbornly opposed or willful.
something that is contrary or opposite:
to prove the contrary of a statement.
either of two contrary things.
Logic. a proposition so related to another proposition that both may not be true though both may be false, as with the propositions “All judges are male” and “No judges are male.”.
in opposition; oppositely; counter:
to act contrary to one’s own principles.
by contraries, contrary to expectation.
on the contrary,

in opposition to what has been stated.
from another point of view:
On the contrary, there may be some who would agree with you.

to the contrary,

to the opposite effect:
I believe he is innocent, whatever they may say to the contrary.
to a different effect.

Contemporary Examples

The Democrats’ Tea Glee Lloyd Grove September 15, 2010
And now, “Optimal” Is a Bad Word Michael Tomasky October 18, 2012
Up to Speed: What’s Happening to Nigeria’s Kidnapped Girls? Nina Strochlic May 6, 2014
A Centrist Government in Israel? Brent E. Sasley January 17, 2013
Newt Gingrich Calls for Universal Right to Bear Arms at NRA Forum Michael Ames April 12, 2012

Historical Examples

The History of Louisiana Le Page Du Pratz
Explorations in Australia John Forrest
The Valet’s Tragedy and Other Stories Andrew Lang
Clarissa, Volume 1 (of 9) Samuel Richardson
The Correspondence of Madame, Princess Palatine, Mother of the Regent; of Marie-Adlade de Savoie, Duchesse de Bourgogne; and of Madame de Maintenon, in Relation to Saint-Cyr Charlotte-Elisabeth, duchesse d Orlans; Marie Adelaide, of Savoy, Duchess of Burgundy; and Madame de Maintenon

adjective
opposed in nature, position, etc: contrary ideas
(kənˈtrɛərɪ). perverse; obstinate
(esp of wind) adverse; unfavourable
(of plant parts) situated at right angles to each other
(logic) (of a pair of propositions) related so that they cannot both be true at once, although they may both be false together Compare subcontrary (sense 2), contradictory (sense 3)
noun (pl) -ries
the exact opposite (esp in the phrase to the contrary)
on the contrary, quite the reverse; not at all
either of two exactly opposite objects, facts, or qualities
(logic) a statement that cannot be true when a given statement is true
adverb (usually foll by to)
in an opposite or unexpected way: contrary to usual belief
in conflict (with) or contravention (of): contrary to nature
adj.

If we take the statement All men are mortal, its contrary is Not all men are mortal, its converse is All mortal beings are men, & its opposite is No men are mortal. The contrary, however, does not exclude the opposite, but includes it as its most extreme form. Thus This is white has only one opposite, This is black, but many contraries, as This is not white, This is coloured, This is dirty, This is black; & whether the last form is called the contrary, or more emphatically the opposite, is usually indifferent. But to apply the opposite to a mere contrary (e.g. to I did not hit him in relation to I hit him, which has no opposite), or to the converse (e.g. to He hit me in relation to I hit him, to which it is neither contrary nor opposite), is a looseness that may easily result in misunderstanding; the temptation to go wrong is intelligible when it is remembered that with certain types of sentence (A exceeds B) the converse & the opposite are identical (B exceeds A). [Fowler]

As a noun from late 13c. Related: Contrarily; contrariwise.
see:

on the contrary
to the contrary