We introduce and study a natural multispecies variant of the inhomogeneous PushTASEP with site-dependent rates on the finite ring. We show that the stationary distribution of this process is proportional to the ASEP polynomials at $q = 1$ and $t = 0$. This is done by constructing a multiline process which projects to the multispecies PushTASEP, and identifying its stationary distribution using time-reversal arguments. We also study symmetry properties of the process under interchange of the rates associated to the sites. These results hold not just for events depending on the configuration at a single time in equilibrium, but also for systems out of equilibrium and for events depending on the path of the process over time. Lastly, we give explicit formulas for nearest-neighbour two-point correlations in terms of Schur functions.