What does the value of political equality imply for the institutional design of democracies? The existing normative literature highlights the importance of proportional representation and legislative majority rule, but neglects the choice of an executive format. This paper explores two potential egalitarian trade-offs in this choice. First, while presidential systems tend to achieve too little bundling of separable decision-making issues (within political parties), parliamentary systems often tend towards too much bundling (between political parties), thus establishing informal veto positions in the democratic process. This is a trade-off between the ‘adversarial’ and ‘deliberative’ aspects of equality. Second, there is a trade-off between ‘horizontal’ and ‘vertical’ equality. Neither pure presidentialism nor pure parliamentarism may be able to maximise both dimensions of equality simultaneously. The paper argues that certain hybrids between parliamentarism and presidentialism have the potential to mitigate both trade-offs. These hybrids establish power separation between the executive and legislature without allowing for popular executive elections. The argument also has potential implications for the democratisation of the European Union.