Integrity not only means that your actions are aligned with your words, it also implies that both are ethically bound. Most importantly, it implies 360-degree accountability, meaning that you are accountable above, below, and sideways, in the organization.
That's not to say you can't make a final decision (and indeed you must), but you need to observe, listen, and consider all angles and impacts before making what you feel is the right decision for the organization. And, for real success, you need to do it quickly. This is done easier in an environment of decentralized-decision making, operating under a set of basic guidelines and high level objectives.
When operating in a more autocratic environment, it can be challenging, especially if the orders conflict with your better instinct.
Here's a great quote I came across in a Tom Peters presentation. The quote comes from Colonel John Boyd, the legendary fighter pilot who founded the concept of OODA loops (Observe, Orient, Decide, and Act), which also helped organizations like Toyota become so successful.
“If your boss demands loyalty, give him integrity. But if he demands integrity, give him loyalty.”
This is a motto we should all carry with us.