For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness
of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals,
and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and
category ideals I0 and I1 on the real line ℝ are partition (or, equivalently, if just I0 is partition), then not
only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups
coinciding, respectively, with the symmetry groups of the polars I⊥0 and I⊥1. The measure and category
ideals on ℝ (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this
case their polars are, respectively, the ideals generated by c-Sierpiński and c-Lusin sets. It is well known
that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the
isomorphism of their symmetry groups is likewise unprovable.