  By a symmetry of the Julia set of a polynomial, also referred as polynomial
Julia set, we mean an Euclidean isometry preserving the Julia set. Each such
symmetry is in fact a rotation about the centroid of the polynomial. In this
article, a survey of the symmetries of polynomial Julia sets is made. Then the
Euclidean isometries preserving the Julia set of rational maps are considered.
A rotation preserving the Julia set of a rational map is called a rotational
symmetry of its Julia set. A sufficient condition is provided for a rational
map to have rotational symmetries whenever the rational map has an exceptional
point. Two classes of rational maps are provided whose Julia sets have
rotational symmetries of finite orders. Using this, it is proved that $
z\mapsto \mu z$ where $\mu^{m+n}=1$ is a rotational symmetry of the McMullen
map $ z^m+\frac{\lambda}{z^n}$ for all $m,n$ with $m\geq 2$ and $\lambda \in
\mathbb{C}\setminus \{0\}$. Assuming that a normalized polynomial has a simple
root at the origin, it is shown that the groups of the rotational symmetries of
the polynmial coincide with that of its Newton's method and Chebyshev's method.
