Consider Euler equation for continuum body:
$$\frac{\partial u^i}{\partial t}+\mathbf u\cdot \nabla u^i=- \frac{1}{\rho} \frac{\partial p}{\partial x^i} $$
where $\rho$ is the mass density, $p$ is the pressure and $\mathbf u (t, \mathbf x)$ is the velocity field. Under the effect of a boost $(t=t',R=1)$:
$$\mathbf u'(t,\mathbf x')= \mathbf u (t,\mathbf x' + \mathbf v t)- \mathbf v=\mathbf u (t,\mathbf x)- \mathbf v$$
(here $\mathbf v$ is the velocity of reference frame)
we note that if we put in a natural way $\rho '(t,\mathbf x')=\rho (t, \mathbf x)$ and $p'(t,\mathbf x')=p (t, \mathbf x)$ we obtain that Euler equation is Galilean covariant BUT is not invariant for a time inversion (i.e. $t \rightarrow -t$).
My question is: which transformations of $\rho$ and $p$ do we have to use in order to obtain Euler equation that is invariant for time-inversion, without losing Galilean covariance?