Us versus them. There are two sides, and the speaker’s group is one of them.
The competition. This is the rival company or team, as in “What is the competition doing?”
Not invented here. This is the habit of rejecting an idea because it came from another team.
Circle the wagons. The group closes ranks against a threat from outside.
Bulletin-board material. This is a remark by an opponent that a team pins up and uses as motivation.
Friendly rivalry. This is competition between groups that still like each other.
Crosstown rivals. These are two teams from the same city.