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We show that the dominant Gamow-Teller part, $M^{0\nu}_{GT}$, of the nuclearmatrix element governing the neutrinoless $\beta\beta$ decay is related to thematrix element $M^{2\nu}_{cl}$ governing the allowed two-neutrino $\beta\beta$decay. That relation is revealed when these matrix elements are expressed asfunctions of the relative distance $r$ between the pair of neutrons that aretransformed into a pair of protons in the $\beta\beta$ decay. Analyzing thisrelation allows us to understand the contrasting behavior of these matrixelements when $A$ and $Z$ is changed; while $M^{0\nu}_{GT}$ changes slowly andsmoothly, $M^{2\nu}$ has pronounced shell effects. We also discuss thepossibility of phenomenological determination of the  $M^{2\nu}_{cl}$ and from them of the $M^{0\nu}_{GT}$ values from theexperimental study of the $\beta^{\pm}$ strength functions.

Relation between the $0\nu\beta\beta$ and $2\nu\beta\beta$ nuclear matrix elements revisited