Moduli spaces of Abelian and quadratic differentials are stratified
by multiplicities of zeroes; connected components of the strata correspond
to ergodic components of the Teichmüller geodesic flow. It is known that the
strata are not necessarily connected; the connected components were recently
classified by M. Kontsevich and the author and by E. Lanneau. The strata can
be also viewed as families of flat metrics with conical singularities and with
$\mathbb Z$/$2 \mathbb Z$-holonomy.
 
For every connected component of each stratum of Abelian and quadratic
differentials we construct an explicit representative which is a Jenkins–Strebel
differential with a single cylinder. By an elementary variation of this construction we represent almost every Abelian (quadratic) differential in the corresponding connected component of the stratum as a polygon with identified
pairs of edges, where combinatorics of identifications is explicitly described.
 
Specifically, the combinatorics is expressed in terms of a generalized permutation. For any component of any stratum of Abelian and quadratic differentials we construct a generalized permutation in the corresponding extended
Rauzy class.