The inhomogeneous multispecies PushTASEP is an interacting particle system with multiple species of particles on a finite ring where the hopping rates are site-dependent. (The homogeneous variant on \( \mathbb Z \) is also known as the Hammersley-Aldous-Diaconis process.) In its simplest variant with a single species, a particle at a given site will hop to the first available site clockwise. We show that the partition function of this process is intimately related to the classical Macdonald polynomial. We also show that large families of observables satisfy the property of interchangeability, namely they have the same distribution even in finite time when the hopping rates are permuted. For some reason, Schur polynomials seem to appear as expectations in the stationary distribution of important observables. This is joint work with James Martin and Lauren Williams and based on the preprints arXiv:2310.09740 and arXiv:2403.10485.