I feel like trading spaces every month or so with a different employee for a brief period of time would be a really, really positive exercise. If you could arrange it so you can mitigate business disruptions while people fumble around (maybe do whatever is typically done with new hires), I have a feeling the mutual appreciation for what other people contribute would lead to better cooperation.
I know there are lots of TV shows and movies about this, so it’s not a novel idea or anything, and I understand the potential for it to backfire, but I can’t overstate the cooperative benefits of gratitude if you really understand and appreciate what it is everyone is contributing.