Intransitive communities, those in which species' abilities cannot be ranked in a hierarchy, have been the focus of theoretical and empirical research, as intransitivity could help explain the maintenance of biodiversity. Here we show that models for intransitive competition embedding slightly different interaction rules can produce opposite patterns. In particular, we find that interactions in which an individual can be outcompeted by its neighbors, but cannot outcompete its neighbors, produce negative frequency dependence that, in turn, promotes coexistence. Whenever the interaction rule is modified toward symmetry (the individual and the neighbors can outcompete each other) the negative frequency dependence vanishes, producing different coexistence levels. Macroscopically, we find that asymmetric interactions yield highest biodiversity if species compete globally, while symmetric interactions favor highest biodiversity if competition takes place locally.