We prove that the acyclic reorientation poset of a directed acyclic graph $D$ is a lattice if and only if the transitive reduction of any induced subgraph of $D$ is a forest. We then show that the acyclic reorientation lattice is always congruence normal, semidistributive (thus congruence uniform) if and only if $D$ is filled, and distributive if and only if $D$ is a forest. When the acyclic reorientation lattice is semidistributive, we introduce the ropes of $D$ that encode the join

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... s acyclic reorientations and exploit this combinatorial model in three directions. First, we describe the canonical join and meet representations of acyclic reorientations in terms of non-crossing rope diagrams. Second, we describe the congruences of the acyclic reorientation lattice in terms of lower ideals of a natural subrope order. Third, we use Minkowski sums of shard polytopes of ropes to construct a quotientope for any congruence of the acyclic reorientation lattice.