I work hard to be trustworthy. Partly, this is because it's the right thing to do (darn my inconvenient morality!), but partly this is because it has real business benefits.
When most of us think about the value of being trustworthy, we think that the key driver of value is that people believe us.
In other words, trust is valuable because it allows you to persuade others.
I think this is naive. First of all, there are many persuasion techniques that rely on deception, rather than trustworthiness. Just think of all the pick up artists that have arisen over the past decade. (Or don't. Ugh.) Second, if you earn people's trust through your honesty, then you're not going to try to persuade others to do the wrong thing. If you're trying to get people to do the right thing, presenting clear evidence is more important than trustworthiness.
The real business benefit of being trusted comes from speed. If you have to prove everything to everyone, you might end up persuading everyone in the end, but at the cost of a lot of time and effort.
If, on the other hand, you're considered trustworthy, people are willing to act on your recommendation, knowing that while they are going to check the evidence later, it will almost certainly support that recommendation.
The end result is the same, but the speed with which you reach it is much greater.
In the startup world, small teams are nimble, not just because of size, but because of the high degree of trust between founders. This is also why a team of founders that don't trust each other are likely to go nowhere fast (with the apparent exception of Facebook, of course).