  We analyze nuclear matrix elements (NME) of neutrinoless double beta decay
calculated for the Cadmium isotopes. Energy density functional methods
including beyond mean field effects such as symmetry restoration and shape
mixing are used. Strong shell effects are found associated to the underlying
nuclear structure of the initial and final nuclei. Furthermore, we show that
NME for two-neutrino double beta decay evaluated in the closure approximation,
$M^{2\nu}_{\mathrm{cl}}$, display a constant proportionality with respect to
the Gamow-Teller part of the neutrinoless NME, $M^{0\nu}_{\mathrm{GT}}$. This
opens the possibility of determining the $M^{0\nu}_{\mathrm{GT}}$ matrix
elements from $\beta^{\mp}$ Gamow-Teller strength functions. Finally, the
interconnected role of deformation, pairing, configuration mixing and shell
effects in the NMEs is discussed.
