  The moduli space of planar polygons with generic side lengths is a closed,
smooth manifold. Mapping a polygon to its reflected image across the $X$-axis
defines a fixed-point-free involution on these moduli spaces, making them into
free $\mathbb{Z}_2$-spaces. There are some important numerical parameters
associated with free $\mathbb{Z}_2$-spaces, like index and coindex. In this
paper, we compute these parameters for some moduli spaces of polygons. We also
determine for which of these spaces a generalized version of the Borsuk-Ulam
theorem hold. Moreover, we obtain a formula for the Stiefel-Whitney height in
terms of the the genetic code, a combinatorial data associated with side
lengths.
