When it comes to working within an office, what is more important: your integrity or the organization's?

Both are very important, you cannot work without integrity, this will add lots of personal values that will give you self-confidence.
Both integrities goes along with each other.

I believed a good organization to work should not sacrifice the integrity of its employee.
It reflects the organization itself; it's a matter of doing business within acceptable and universal standards of doing business.
It's the organizations social responsibility to upheld the integrity of its employee!

my integrity inline with the organizations integrity

Always we should strictly adhere to: standards professional ethics, which will retain rights for all Business parties.

your integrity is in your control and important is one's own integrity and once you show integrity, the organization should respect your integrity in form of reward.