Abstract In this work, we study (anti-)self duality conditions in unconventional conformal supersymmetry. We focus on a theory constructed in a Townsend-MacDowell-Mansouri form for an SU(2, 2|N) gauge connection with matter fields in the adjoint representation. We find bosonic solutions that correspond to analytic gravitational instantons with nontrivial torsion. These configurations can be regarded as the torsional generalization of the Taub-NUT/Bolt-AdS and Eguchi-Hanson metric and they are (anti-)self-dual with respect to a generalized dual operator. We explore their global properties and show that they saturate a BPS bound.