An interfolding machine includes a first folding roll with a series of the gripper assemblies and a series of tucker assemblies uniformly and alternately spaced to interact with a series of gripper and tucker assemblies of an adjacent second folding roll. The series of alternately spaced gripper and tucker assemblies interact to grip, carry, and release a sheet of material in a manner so as to generate an interfolded stack of sheets. Each tucker assembly includes an outwardly biased tucker element that is operable to self-center in a slot formed in the folding roll. The tucker assembly further includes a roller mounted by a pin to the tucker element such that the tucker element rolls along a surface defined by the slot. Another embodiment of the tucker assembly includes a pivot spring and bumper to self-center the tucker element in the slot.