Are opposite-order equilibria impossible?

An opposite-order platform profile occurs when parties choose platforms in the opposite order from their ideal points. Such an outcome undermines the usual ideological interpretation of party competition: the party identified as left chooses the right-hand platform, while the party identified as right chooses the left-hand platform. We study a two-party ordinal spatial voting model in which voter and party preferences are single-peaked. Under full turnout, we show that single-peakedness alone rules out opposite-order equilibria. With endogenous abstention, we identify an additional ordinal condition: parties agree on how to rank deviations on either side of their ideal points. We call this cross-side agreement. Cardinal utilities that are commonly used in the literature, like symmetric single-peaked utilities and common translated utility shapes, satisfy this condition. Under single-peakedness and cross-side agreement, opposite-order equilibria are impossible even with abstention. The assumption is tight: we construct an example admitting a unique pure-strategy Nash equilibrium with opposite order in which cross-side agreement fails.Length: 17pages