On 2-star-permutability in regular multi-pointed categories

2-star-permutable categories were introduced in a joint work with Janelidze and Ursini as a common generalization of regular Mal'tsev categories and of normal subtractive categories. In the present paper, we first characterize these categories in terms of what we call star-regular pushouts. We then show that the 3 x 3 Lemma characterizing normal subtractive categories and the Cuboid Lemma characterizing regular Mal'tsev categories are special instances of a more general homological lemma for star-exact sequences. We prove that 2-star-permutability is equivalent to the validity of this lemma for a star-regular category.