Geometric combinatorics is the art of studying discrete structures by way of geometry. True gems in this area are Stanley’s “two poset polytopes”. The order polytope and the chain polytope reflect much of the combinatorics of partially ordered sets (or, posets) in their boundary structures, their volumes, and their Ehrhart polynomials. In this talk I will discuss four more such polytopes associated to partial orders with applications to permutation statistics, increasing/alternating sequences, and valuations on distributive lattices. On the geometric side, these polytopes make interesting connections to anti-blocking polytopes from combinatorial optimization, compressed and equidissectable polytopes from discrete geometry, and arrangements of tropical min- and max-hyperplanes.