  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.
