Summary: Two different characterizations of self-dual aggregation operators are available in the literature: one based on \({\mathcal C}(x,y)= x/(x+1-y)\) and one based on the arithmetic mean. Both approaches construct a self-dual aggregation operator by combining an aggregation operator with its dual. In this paper, we fit these approaches into a more general framework and characterize \(N\)-invariant aggregation operators, with \(N\) an involutive negator. Various binary aggregation operators, fulfilling some kind of symmetry w.r.t. \(N\) and with a sufficiently large range, can be used to combine an aggregation operator and its dual into an \(N\)-invariant aggregation operator. Moreover, using aggregation operators to construct \(N\)-invariant aggregation operators seems rather restrictive. It suffices to consider \(n\)-ary operators fulfilling some weaker conditions. Special attention is drawn to the equivalence classes that arise as several of these \(n\)-ary operators can yield the same \(N\)-invariant aggregation operator.