  The hypersimplex $\Delta_{k+1,n}$ is the image of the positive Grassmannian
$Gr^{\geq 0}_{k+1,n}$ under the moment map. It is a polytope of dimension $n-1$
in $\mathbb{R}^n$. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}(Z)$ is the
projection of the positive Grassmannian $Gr^{\geq 0}_{k,n}$ into $Gr_{k,k+2}$
under a map $\tilde{Z}$ induced by a matrix $Z\in \text{Mat}_{n,k+2}^{>0}$.
Introduced in the context of scattering amplitudes, it is not a polytope, and
has dimension $2k$. Nevertheless, there seem to be remarkable connections
between these two objects via T-duality, as was first noted by
Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented
matroid theory, total positivity, and the geometry of the hypersimplex and
positroid polytopes to obtain a deeper understanding of the amplituhedron. We
show that the inequalities cutting out positroid polytopes -- images of
positroid cells of $Gr^{\geq 0}_{k+1,n}$ under the moment map -- translate into
sign conditions characterizing the T-dual Grasstopes -- images of positroid
cells of $Gr^{\geq 0}_{k,n}$ under $\tilde{Z}$. Moreover, we subdivide the
amplituhedron into chambers, just as the hypersimplex can be subdivided into
simplices, with both chambers and simplices enumerated by the Eulerian numbers.
We prove the main conjecture of (LPW): a collection of positroid polytopes is a
triangulation of $\Delta_{k+1, n}$ if and only if the collection of T-dual
Grasstopes is a triangulation of $\mathcal{A}_{n,k,2}(Z)$ for all $Z$.
Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip
characterization of $\mathcal{A}_{n,k,2}(Z)$, and
Lukowski--Parisi--Spradlin--Volovich's conjectures on $m=2$ cluster adjacency
and on generalized triangles (images of $2k$-dimensional positroid cells which
map injectively into $\mathcal{A}_{n,k,2}(Z)$). Finally, we introduce new
cluster structures in the amplituhedron.
