  Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full
isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat
manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$
are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$
are representation equivalent, that is, the right regular representations
$L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ are unitarily
equivalent.
