With Great Power Comes Great duty.
Lawful Neutral characters believe the best way is to have a specific, strict code of conduct, whether self-imposed or codified as a law. Their first impulse when making a moral decision is to refer back to this code; those with externally imposed systems (codes of laws, hierarchies, etc.) will try to work within the system when those systems go wrong. They will refuse to break this code even if good could be done by breaking it.