All 5 Uses of

contrary
in
Washington Square, by Henry James

On the contrary, for fear of being unjust to her, he did his duty with exemplary zeal, and recognised that she was a faithful and affectionate child.†
Chpt 2 *

on the contrary = an expression used to intensify denial of an idea

It is very possible that his motives are pure; I should be very sorry to take the contrary for granted.†
Chpt 6 *

This question was not reassuring; it seemed to Catherine, on the contrary, to suggest possibilities which made her feel sick.†
Chpt 11

on the contrary = an expression used to intensify denial of an idea

It was not that she was glad of his embarrassment; on the contrary, it excited all her own liabilities of this kind, and gave her great pain.†
Chpt 35

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

More rarely, on the contrary is used to oppose an idea contained in what was just said rather than to oppose what was literally said. Here is an example: "I will not pay you for the work. On the contrary, I may sue you for damages." Suing for damages does not oppose not paying, but it does oppose the idea of paying.
(5) (to the contrary as in: to the contrary)
with an opposite or different effect; or something with an opposite or different effect