According to this chart, DCFLs are closed under reversal.
However, I am not convinced as the intuitive proof (reversing the arrows of the controlling finite state machine and switching the pushes and pops) for this seems to depend on non-determinism in choosing the null transition to take from the initial state (since the new initial state would contain a null transition to all the old final states).
This would make the "reverse PDA" of a DPDA non-deterministic whenever there is more than a single final state in the original DPDA.
What is the fallacy in my argument? Or is there another way to go about proving this?