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Intimate relation between the Gamow-Teller part of the matrix element$M^{0\nu}_\mathrm{GT}$ and the $2\nu\beta\beta$ closure matrix element$M^{2\nu}_\mathrm{cl}$ is explained and explored. If the corresponding radialdependence $C^{2\nu}_\mathrm{cl}(r)$ would be known, $M^{0\nu}$ correspondingto any mechanism responsible for the $0\nu\beta\beta$ decay can be obtained asa simple integral. However, the $M^{2\nu}_\mathrm{cl}$ values sensitivelydepend on the properties of higher lying $1^+$ states in the intermediateodd-odd nuclei. We show that the $\beta^-$ and $\beta^+$ amplitudes of suchstates typically have opposite relative signs, and their contributions reduceseverally the $M^{2\nu}_\mathrm{cl}$ values. Vanishing values of$M^{2\nu}_\mathrm{cl}$ are signs of a partial restoration of the spin-isospin$\mathrm{SU}(4)$ symmetry. We suggest that demanding that$M^{2\nu}_\mathrm{cl}$ = 0 is a sensible way, within the method of theQuasi-particle Random Phase Approximation (QRPA), of determining the amount ofrenormalization of isoscalar particle-particle interaction strength$g^{T=0}_{pp}$. Using such prescription, the matrix elements $M^{0\nu}$ areevaluated; their values are not very different ($\le$ 20\%) from the usual QRPAvalues when $g^{T=0}_{pp}$ is related to the known $2\nu\beta\beta$ half-lives.

$0\nu\beta\beta$ nuclear matrix elements, neutrino potentials and $\mathrm{SU}(4)$ symmetry