All 4 Uses

contrary
in
Henry VIII
(Auto-generated)

The King's attorney on the contrary urged on the examinations, proofs, confessions of...
Scene 2.1 *

on the contrary = an expression used to intensify denial of an idea
In the divorce his contrary proceedings Are all unfolded; wherein he appears As I would wish mine enemy.†
Scene 3.2 *
The honour of it Does pay the act of it, as i' the contrary, The foulness is the punishment.†
Scene 3.2
If they shall chance, In charging you with matters, to commit you, The best persuasions to the contrary Fail not to use, and with what vehemency The occasion shall instruct you.†
Scene 5.1 *

Definitions:

(1) (contrary as in: a contrary idea)
different (perhaps opposite or mutually exclusive)

In formal logic, contrary propositions or contrary arguments describe two things that cannot both be true. They might be described as mutually exclusive. For example, a pet cannot be both a cat and a dog. It doesn't need to be either, but if it is one, it cannot be the other. So the statement that a pet is a cat and the statement that the same pet is a dog, are contrary statements.
(2) (contrary as in: a contrary personality)
disagreeable -- typically in reference to someone's personality
(3) (contrary to as in: contrary to)
in opposition to

You could also think of this as meaning, despite, or in spite of, or in conflict with; but since a common meaning of contrary is opposite, you many find it easiest to think of contrary to as in opposition to.
(4) (on the contrary as in: on the contrary)
an expression used to intensify denial of an idea
(5) (to the contrary as in: to the contrary)
with an opposite or different effect; or something with an opposite or different effect
(6) (meaning too rare to warrant focus)