  A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$
in $\mathbb{R}^d$ and the collection of all such forms the configuration space
$\mathcal{C}(\Gamma)$ of $\Gamma$. We here introduce the notion of the
$symmetric configuration space$ of a linkage, in which we identify
configurations which are geometrically indistinguishable. We show that the
symmetric configuration space of a planar polygon has a regular cell structure,
provide some principles for calculating this structure, and give a complete
description of the symmetric configuration space of all quadrilaterals and of
the equilateral pentagon.
