If $L$ is regular, must the language $L_1 = \{w : w^Rw \in L\}$ be regular, or may it be non-regular?
It will be regular.


   Let $M$ be the syntactic monoid of $L$ and consider the monoid $M^{\operatorname{op}} \times M$ (where $M^{\operatorname{op}}$ is $M$ with multiplication flipped).