All 5 Uses of

contrary
in
Atlas Shrugged

I think it's contrary to the intent of the Anti-dog-cat-dog Rule," he said angrily.†
Chpt 1.7

contrary to = in opposition to

Contrary to a business firm, the State Science Institute had no trouble in obtaining funds for the Project.†
Chpt 3.3

She stared in astonishment at a paragraph in a syndicated column from New York, which stated over emphatically that Mr. James Taggart wished it to be known that his sister had died in an airplane crash, any unpatriotic rumors to the contrary notwithstanding.†
Chpt 3.3 *

They proclaim that every man born is entitled to exist without labor and, the laws of reality to the contrary notwithstanding, is entitled to receive his 'minimum sustenance'—his food, his clothes, his shelter-with no effort on his part, as his due and his birthright.†
Chpt 3.7

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