  We uncover a hidden Gaussian ensemble inside each of the three circular
ensembles of random matrices, which provide novel diagrammatic rules for the
calculation of moments. The matrices involved are generic complex for
$\beta=2$, complex symmetric for $\beta=1$ and complex self-dual for $\beta=4$,
and their dimension must be set to $1-2/\beta$. As an application, we compute
moments of traces of submatrices.
