  A DNA string is a Watson-Crick (WK) palindrome when the complement of its
reverse is equal to itself. The Watson-Crick mapping $\theta$ is an involution
that is also an antimorphism. $\theta$-conjugates of a word is a generalisation
of conjugates of a word that incorporates the notion of WK-involution $\theta$.
In this paper, we study the distribution of palindromes and Watson-Crick
palindromes, also known as $\theta$-palindromes among both the set of
conjugates and $\theta$-conjugates of a word $w$. We also consider some general
properties of the set $C_{\theta}(w)$, i.e., the set of $\theta$-conjugates of
a word $w$, and characterize words $w$ such that $|C_{\theta}(w)|=|w|+1$, i.e.,
with the maximum number of elements in $C_{\theta}(w)$. We also find the
structure of words that have at least one (WK)-palindrome in $C_{\theta}(w)$.
