In Marchioro and Pulvirenti's book Mathematical Theory of Incompressible Nonviscous Fluids, the proof of global well-posedness of the 2D Euler equation in a bounded domain $\Omega\subset\mathbb R^2$ works by approximating the flow $\Phi^n_t(x)$ of the fluid given initial vorticity $\omega_0\in L^\infty(\Omega)$, and showing that $\Phi^n\to\Phi$ for some $\Phi$, with convergence in $L^\infty([0,T];L^1(\Omega))$, and similar arguments show convergence $(\Phi^n)^{-1}\to\tilde\Phi$ for some $\tilde\Phi$. Implicitly, the authors assume that $\Phi_t^{-1}=\tilde \Phi_t$ for all $t$, but I don't see why this must hold.
If it is of any use, I have found that the $\Phi^n$ and their inverses must all be incompressible flows, which are differentiable in time, and $C^{0,s}$ in space for some $s>0$.