  Neutrinoless double beta ($0\nu\beta\beta$) decay can in general produce
electrons of either chirality, in contrast with the minimal Standard Model (SM)
extension with only the addition of the Weinberg operator, which predicts two
left-handed electrons in the final state. We classify the lepton number
violating (LNV) effective operators with two leptons of either chirality but no
quarks, ordered according to the magnitude of their contribution to \znbb
decay. We point out that, for each of the three chirality assignments, $e_Le_L,
e_Le_R$ and $e_Re_R$, there is only one LNV operator of the corresponding type
to lowest order, and these have dimensions 5, 7 and 9, respectively. Neutrino
masses are always induced by these extra operators but can be delayed to one or
two loops, depending on the number of RH leptons entering in the operator.
Then, the comparison of the $0\nu\beta\beta$ decay rate and neutrino masses
should indicate the effective scenario at work, which confronted with the LHC
searches should also eventually decide on the specific model elected by nature.
We also list the SM additions generating these operators upon integration of
the heavy modes, and discuss simple realistic examples of renormalizable
theories for each case.
