Are DCFLs closed under reversal?
I looked up Hopcroft and Ullman 1979 and it say on page 281 that it is not closed under reversal. But I  found no proof in my very fast look at the relevant chapter.
Searching the web does also give a negative answer, with counter example, on stackoverflow by a member of CS (notation adapted):

$(a+b+c)^*WcW^R$, where $W \in (a+b)^+$; this is non-deterministic because you don't know where the $WcW$ bit starts.
$W^RcW(a+b+c)^*$, where $W \in (a+b)^+$; this is deterministic because you can write a deterministic PDA to accept simple palindromes of the form $W^RcW$ and modify the accepting state to loop to itself on any of $a$, $b$ and $c$.
The trick here is that PDAs have to read input from left to right.