All 6 Uses of

contrary

in

Howards End

On the contrary—†
Part 2 *

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

It was not legal; it had been written in illness, and under the spell of a sudden friendship; it was contrary to the dead woman's intentions in the past, contrary to her very nature, so far as that nature was understood by them.†
Part 11 *

contrary to = in opposition to

It was not legal; it had been written in illness, and under the spell of a sudden friendship; it was contrary to the dead woman's intentions in the past, contrary to her very nature, so far as that nature was understood by them.†
Part 11

"I have a message, dear Helen," etc., and so she had, but had been very nice when Helen laughed; quite understood—a forest too solitary and damp—quite agreed, but Herr Forstmeister believed he had assurance to the contrary.†
Part 12 *

England was alive, throbbing through all her estuaries, crying for joy through the mouths of all her gulls, and the north wind, with contrary motion, blew stronger against her rising seas.†
Part 19 *

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
(6) (meaning too rare to warrant focus)