Finding examples of languages that are "anti-palindromic"
One example will be $L = \{ x\ \ |\ \   binary(x) &lt; binary(x^R), x\in [0,1]^*\}$. 

And yet another example $L' = \{ x\ \ |\ \   binary(x) &gt; binary(x^R), x\in [0,1]^*\}$. 

The idea is, if $x \neq x^R$ you make  a rule to choose only one of them. You need to choose the rule such that palindromes should be rejected ($f(x) &lt; f(x^R)$, for palindromes you must have $f(x)= f(x^R)$).You can also change the alphabet, I took binary alphabet just to get a quick answer.

$L$ and $L'$ above are not regular. And every anti-palindromic language will be non-regular and can be as bad as a non-RE language. Examble of an undecidable language: $L=\{x\ \ |\ \ $such that $ binary(x)&lt;binary(x^R)$ if both $x$ and $x^R$ $\not\in$ Halt or both $x$ and $x^R$ $\in$ Halt, otherwise if $x \in$ Halt$\}$

Klaus Draeger explained in the comment below that anti-palindromic language given at the beginning of the answer is context-free: $L=\{x0y1x^R\ |\ x,y\in\{0,1\}^*\}$