First half of context-free palindromes
Yes.



Sándor Horváth, J. Karhumäki and H. C. M. Kleijn characterized the context-free languages consisting only of palindromes in their paper Results Concerning Palindromicity, 1987. 

Theorem. A context-free language $L\subseteq\Sigma^*$ is palindromic, iff it is of the form 
$$ L= \bigcup_{x\in\,\Sigma\cup\{\lambda\}}\{uxu^R : u\in L(x)\}$$

where the $L(x)$ is a regular language uniquely determined by $L$ for all $x \in\Sigma\cup\{\lambda\}$.



In particular, the theorem tells that a context-free languages that consists of only palindromes of even length must be of the form $\{uu^R : u\in L\}$ for some regular language $L$.