Down-the-middle isometries for the Airy and KPZ sheets

We find a new coupling between the Airy sheet $\mathcal{S}$ and the Airy line ensemble $\{{\mathcal{A}}_n{\}}_{n\in \mathbb{N}}$, which we call the down-the-middle isometry. In this coupling, the Airy sheet is encoded by last passage values that start at ${\mathcal{A}}_k(0)$ for some $k\in \mathbb{N}$ and end on the top line ${\mathcal{A}}_1$. Using this coupling, we give a new construction of the (extended) Airy sheet as a scaling limit of Brownian last passage percolation and study coalescence of geodesics in the directed landscape. This coupling extends to discrete polymers and last passage percolation models, as well as to the KPZ line ensemble and KPZ sheet.