The Jefferson Memorial, Washington
expressing Jefferson's sentiment on the freedom of religion
To be sure, there's a place for PMO governance, but culture shouldn't need a lot of it. Culture should more or less be inherited from the business -- sometimes from the customer if you are a contractor -- and not require constant attention from the PMO.
But... if you find yourself writing a lot of behavior rules, waxing on at PMO meetings about the culture, then you're likely treating symptoms and not root cause. And, the root cause is probably back in the business.
Culture is a lagging indicator -- but of what? It's a lagging indicator of how well beliefs, etc have been internalized rather than just spoken about. And the internalization is usually led from the top, inherited by all below, and has it's root in the C-Suite
Check out these books I've written in the library at Square Peg Consulting