There's an integrity test that never fails, and you're not the one who runs it: your team runs it every day. It's this: when you announce something, do people act, or do they wait to see what you do?
If they wait, they already know something you may not have wanted to see: that what you say and what you do aren't always the same thing.
Integrity is your word being trusted: that you do what you say, walk the talk, and hold the same substance in every room. It doesn't mean saying everything. It doesn't mean being rigid. It doesn't mean practising the cruelty of «total honesty». It means something simpler and harder: that your word is worth the same even when the audience changes.
What covers it is image and convenience: adjusting the message to whoever is listening. One version for the board; another for the committee; in the corridor, perhaps the real one. Each adjustment looks like political intelligence, and some of it is. But the sum has a silent price: people compare versions. They always compare. And the day they discover two speeches, they stop believing both.
Integrity doesn't demand that you tell everything. It demands that you don't tell something else. When you can't share something, you can say: «I can't share this yet.» That builds more trust than a well-wrapped half-truth.
Your exception: remember when you honoured a commitment that had stopped suiting you, or defended to the board the same thing you'd told your team, knowing it would cost you. What happened to your authority? It went up. Comfortable integrity barely counts. The kind that scores is the kind that costs.
The small experiment: for one week, before every important meeting, ask yourself one question: «is what I'm about to say here compatible with what I said there?». When the answer is no, adjust it before walking in, or name openly why there's a difference. You'll feel the cost immediately and the benefit later.
When this competency grows, your power changes source. It stops coming from the title and starts coming from you. A reorganisation can take the title; it can't take your word.
My word is worth the same in every room. It's the cheapest asset to build and the most expensive to recover.