The three-dimensional development of strongly non-cylindrical folds around competent ellipsoidal inclusions in bulk simple shear regimes is studied in experiments and natural examples. Examples from the Continental Allochthonous Terrane of the Bragança Nappe Complex (NE Portugal) show rim folds and sheath folds associated with different parts of rigid ellipsoidal boudins. Experimental work has been carried out with models made from analogue materials (silicone putty and plasticine) and deformed in a simple shear machine. We have considered three different models to simulate natural examples, and the results show that fold morphology depends on the shape of the inclusion and position around the inclusion. Although the bulk deformation regime is layer parallel homogeneous simple shear, we can distinguish local deformation regimes responsible for the folding associated with different parts of the rigid body. Flow and strain patterns must therefore be complex around inclusions or populations of inclusions. Prolate to oblate strain ellipsoids can be expected in different positions close to the rigid inclusion. In our experiments, the competent inclusions do not rotate synthetically with the applied bulk simple shear (e.g. clockwise rotation of inclusion in dextral shear). Instead, they back-rotate, early in the shearing history, and keep this position throughout the experiment. This is the result of the size relationship between the inclusion and the finite width of the shear zone. The fold pattern around rigid inclusions may be used as a shear sense criterion.