  We analyze neutrinoless double $\beta$-decay ($\nbb$-decay) mediated by heavy
particles from the standpoint of effective field theory. We show how symmetries
of the $\nbb$-decay quark operators arising in a given particle physics model
determine the form of the corresponding effective, hadronic operators. We
classify the latter according to their symmetry transformation properties as
well as the order at which they appear in a derivative expansion. We apply this
framework to several particle physics models, including R-parity violating
supersymmetry (RPV SUSY) and the left-right symmetric model (LRSM) with mixing
and a right-handed Majorana neutrino. We show that, in general, the pion
exchange contributions to $\nbb$-decay dominate over the short-range
four-nucleon operators. This confirms previously published RPV SUSY results and
allows us to derive new constraints on the masses in the LRSM. In particular,
we show how a non-zero mixing angle $\zeta$ in the left-right symmetry model
produces a new potentially dominant contribution to $\nbb$-decay that
substantially modifies previous limits on the masses of the right-handed
neutrino and boson stemming from constraints from $\nbb$-decay and vacuum
stability requirements.
