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