Is there a word for one end of a dichotomy?

A common definition of a dichotomy is:

division into two parts

or

mutually exclusive or contradictory groups or entities...

or

A division or contrast between two things...

Parts, groups, entities and things are all fine words, but they're all very broad. Almost anything can be a thing, or an entity. Practically everything is a part of something or a group of other things. None of those terms is in any way specific to a dichotomy.

Is there a term that specifically means one end of a dichotomy, or even something close to that?

If not, what's the closest we can get?

Perhaps something defined in terms of opposition to the other, like antithesis?

The sentence that has sent me in search of this (probably non-existent word) is:

There is a tradeoff between password length, and how often we expect users to change passwords, and we have chosen the wrong end-of-the-dichotomy.

I'd like to find a single word to complete the sentence, if there is one.

Solution 1:

The two opposites distinguished by a dichotomy or antithesis are the poles of that dichotomy or antithesis:

Either of two antithetical ideas, propensities, forces, or positions. (American Heritage)

Either of two mutually exclusive or opposite actions, opinions, etc. (Collins)

For your specific sample sentence, however, the following metaphor might be preferable:

There is a tradeoff between password length, and how often we expect users to change passwords, and we have got the wrong end of the stick on that [or perhaps we have got hold of the wrong end of that stick].

On that metaphoric expression, see this question on this site, or this TFD entry. Both point to its primary meaning as a kind of mis-perception or misunderstanding—that is, an error more theoretical than practical—but that seems to fit the case well enough.

Solution 2:

There is a tradeoff between password length, and how often we expect users to change passwords, and we have chosen the wrong side.

Or if you don't like "side", see synonyms for "side" as related to "angle".