Closure properties of the class of inherently ambiguous CFLs
Reversal

The class of inherently ambiguous context-free languages is closed under reversal (exercise). 

Intersection

The class of inherently ambiguous context-free languages is not closed under intersection. Indeed, take some inherently ambiguous context-free language $L$. Now take $L'$ to be a copy of $L$ which uses a different alphabet. Then $L \cap L' = \emptyset$.

Complementation

The class of inherently ambiguous context-free languages is not closed under complementation. Indeed, consider the Goldstine language
$$ G = \{ a^{n_1} b \cdots a^{n_p} b : p \geq 1 \text{ and } n_i \neq i \text{ for some } i \}. $$
This language is context-free and inherently ambiguous (see slides of Cyril Nicaud). Also,
$$
\overline{G} \cap \Sigma^* b = \{ aba^2\cdots a^p b : p \geq 1 \},
$$
which isn't even context-free.

Union

The class of inherently ambiguous context-free languages is not closed under union. Indeed, the following variant of the Goldstine language is probably also context-free and inherently ambiguous:
$$ G' = \{ a^{n_1} b \cdots a^{n_p} b : p \geq 1 \text{ and } n_i \neq i+1 \text{ for some } i \}. $$
However, $G \cup G' = (a^*b)^+$.

Iteration (Kleene star)

The class of inherently ambiguous context-free languages is not closed under Kleene star. Indeed, consider the following language:
$$
\Omega_3 = \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b \text{ or } |u|_a \neq |u|_c \}.
$$
Here $|u|_a$ is the number of $a$s in $u$. This language is context-free and inherently ambiguous (see the slides of Cyril Nicaud). However, $\Omega_3^* = \{a,b,c\}^*$.

Concatenation

The class of inherently ambiguous context-free languages is not closed under concatenation. Indeed, the following variants of $\Omega_3$ are probably also context-free and inherently ambiguous:
$$
\begin{align*}
\Omega'_3 &amp;= \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b \text{ or } |u|_a \neq |u|_c \text{ or } u = \epsilon \}, \\
\Omega''_3 &amp;= \{ u \in \{a,b,c\}^* : |u|_a \neq |u|_b+1 \text{ or } |u|_a \neq |u|_c+1 \text{ or } u = \epsilon \}.
\end{align*}
$$
However, $\Omega'_3 \Omega''_3 = \{a,b,c\}^*$.