When you bring together unlike elements, you unify them. Political movements can unify people by inspiring them to work towards a shared goal. Likewise, a shared love of a team or band can unify people who otherwise don't have much in common.
The verb unify comes from the Latin prefix uni-, “one,” and the root facere, “to make.” When you unify things, you make them one. Often squabbling people just need someone to point out their similarities in order to unify them. If your classmates are sabotaging each other to try to win the “best-decorated locker” award, you might be able to unify their efforts suggesting a “most awesome class” award instead.