We revisit the well known 'Black Box' theorem establishing a fundamental relation between the amplitude of neutrinoless double beta decay and the effective Majorana neutrino mass. We extend this theorem to the general case of arbitrary lepton number and lepton flavor violating (LFNV) processes and to the three generation Majorana neutrino mass matrix. We demonstrate the existence of a general set of one-to-one correspondence relations between the effective operators generating these processes, and elements of the neutrino mass matrix, such that if one of these two quantities vanishes the other is guaranteed to vanish as well, and moreover, if one of these quantities is non-zero the other is guaranteed to be non-zero. We stress that this statement remains valid even in the presence of arbitrary new physics contributions. As a particularly important example, we then show that neutrino oscillation data imply that neutrinoless double beta decay must occur at a certain non-zero rate.