Envy-free, efficient and strategy-proof division of a heterogeneous good between two agents

We study the division of $[0,1]$ between two agents, each receiving a closed subinterval, under ordinal preferences that need not be additive. A monotonic preference has a unique balanced cut, at which the agent is indifferent between the left and the right piece. On a domain of monotonic preferences, a strategy-proof, envy-free and Pareto efficient rule exists if and only if the balanced cuts of the preferences in the domain take at most two distinct values; when they take exactly two, the divide-and-choose rules are the only such rules. For preferences single-peaked in quantity, which are additive only if monotonic, the three properties are compatible when the agents' most preferred bundles can be assigned simultaneously.