  A binary linear code $C$ is a $\mathbb{Z}_2$-double cyclic code if the set of
coordinates can be partitioned into two subsets such that any cyclic shift of
the coordinates of both subsets leaves invariant the code. These codes can be
identified as submodules of the $\mathbb{Z}_2[x]$-module
$\mathbb{Z}_2[x]/(x^r-1)\times\mathbb{Z}_2[x]/(x^s-1).$ We determine the
structure of $\mathbb{Z}_2$-double cyclic codes giving the generator
polynomials of these codes. The related polynomial representation of
$\mathbb{Z}_2$-double cyclic codes and its duals, and the relations between the
polynomial generators of these codes are studied.
