No enemies
I've sat on just about every side of a construction meeting. I started in the field with tools in my hands. I spent a dozen or so years on the contractor side, running work, then running a division. Learning first hand how a contractor makes money and, more importantly, how they lose it. Then I crossed over to the owner's side and spent a decade there. Same industry, three completely different views of the same project.
I've learned that when a job goes sideways, everybody is dead certain they know who the problem is. Everybody is blaming each other, or maybe they're even blaming themselves, but there is rarely ever any ill intent from anyone. Nobody is trying to sink the project. What you have on every single construction project is a mismatch of goals. The project is born from a need, and then it gets scoped according to the available budget. Some of the potential problems start right there — in that part about the available budget.
So the design team comes in, then the contractor, then you'll have all of the subordinate engineers and trades, throw in some inspectors and whatnot, and you'll find yourself with a huge project team, and none of them necessarily share the same goals as the owner. Everybody has their own schedules and budgets that may or may not have anything to do with that original need. That's just how it is. The only thing they all have in common is they all want to be successful, and that turns out to be a pretty good leverage point.
Your job as the PM is to find some way to let them all be successful. If you can do that, then it's a win for you no matter how many disasters you had to navigate. So the most useful thing you do is just simple translation. It's sitting in the room and saying out loud what each side is actually trying to protect or accomplish until everybody realizes they aren't enemies, they were just misunderstanding each other. Nine times out of ten the "dispute" dissolves the moment people stop defending positions and start talking about what they genuinely need. From there it's a short walk to helping each other get it done.