Integrity

Integrity is the virtue of loyalty in action to one's own convictions and values. Where honesty is the refusal to fake facts, Objectivism treats integrity as the refusal to fake one's own consciousness: to profess one thing and act on another under social or material pressure.

“Integrity is the recognition of the fact that you cannot fake your consciousness.”
~ Ayn Rand, Atlas Shrugged

Why a breach is not a bargain

The temptation integrity addresses always has the same shape. A conviction is inconvenient in some particular case; abandoning it in that case appears to cost nothing, because the conviction itself is left intact for all other cases.

Objectivism denies the arithmetic. A principle is an identification of what a whole class of situations requires, so the only place a principle exists is in the actions taken when it is expensive:

A conviction acted against was not knowledge. If evidence would change one's mind, that is rationality. If a bribe, a job or a room full of disapproval changes one's behaviour without changing the evidence, nothing has been reconsidered — only overridden.
The concession sets the terms of the next one. A man who has established that his convictions yield to pressure has told everyone, himself included, what price to offer.
It is the mind that pays. Once action and judgment are allowed to diverge, judgment stops being the thing that guides action and becomes commentary on it. This is the sense in which one cannot fake one's own consciousness: the faking has to be done by the very faculty being faked.

Rand dramatised the alternative in The Fountainhead. Howard Roark and Peter Keating are given the same career problem repeatedly; the difference between them is not talent but whether the answer is settled by the work or by the audience. See Independence.

Integrity is not stubbornness

The virtue is not the refusal to change one's mind, and it is not a demand to announce every conviction at every opportunity.

Changing a conclusion because of new evidence or a corrected argument is rationality operating exactly as it should.
Declining to volunteer one's views to people who have no claim on them is discretion, not evasion. What integrity forbids is asserting agreement one does not hold.
Working within a system one regards as unjust — paying a tax, obeying a law one would repeal — is not a breach. Objectivism distinguishes between acting on a conviction and being able to escape the consequences of other people's actions. See Context.
Compromise on concretes between people who share a principle is ordinary trade. Compromise on the principle itself is what the virtue rules out; Rand made this the subject of "Doesn't Life Require Compromise?".

Common misunderstandings

That it means never conceding anything. Negotiating a price is not a moral failure. Agreeing that theft is sometimes acceptable to get the price down is.
That it is the same as consistency. A consistently held error is not integrity; it is an error. The virtue presupposes that the convictions were reached by reason.
That it makes one unemployable or friendless. What it rules out is buying acceptance by pretending. Most working relationships never require that, and the ones that do are the ones worth identifying early.
That it applies only to large public stands. Rand's emphasis is on the small, unwitnessed cases, because those are where a policy is actually formed.

Discussion

References

Rand, Ayn. "The Objectivist Ethics", The Virtue of Selfishness. New American Library, 1964.
Rand, Ayn. "Doesn't Life Require Compromise?", The Virtue of Selfishness.
Rand, Ayn. Atlas Shrugged — Galt's speech, on integrity as the refusal to fake one's consciousness.
Peikoff, Leonard. Objectivism: The Philosophy of Ayn Rand, chapter 8.

External links

Integrity at the Ayn Rand Lexicon