All 7 Uses of

contrary
in
The Phantom of the Opera

He was bound to conclude in favor of the madness of the viscount and the accidental death of the elder brother, for lack of evidence to the contrary; but he was nevertheless persuaded that a terrible tragedy had taken place between the two brothers in connection with Christine Daae.†
Chpt Prol. *

Mamma Valerius' hair was now quite white, but her eyes had grown no older; never, on the contrary, had their expression been so bright, so pure, so child-like.†
Chpt 8 *

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

"On the contrary, mademoiselle," said the young man, in a voice which he tried to make firm and brave, but which still trembled, "anything that concerns you interests me to an extent which perhaps you will one day understand.†
Chpt 10

Contrary to what one might think, especially in connection with an opera-house, the "organ" is not a musical instrument.†
Chpt 20 *

contrary to = in opposition to

On the contrary, in my longing to approach the source of that sweet and enticing harmony, I leaned out of my little boat over the water, for there was no doubt in my mind that the singing came from the water itself.†
Chpt 21

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

The heat did not go with the daylight; on the contrary, it was now still hotter under the blue rays of the moon.†
Chpt 24

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