Kresling tube metamaterials are well known to exhibit a chirality-dependent exotic mechanical feature: a shortening or lengthening in the direction of the tube’s axis produces a relative rotation of the two polygonal bases of the tube. This property can be easily grasped by fabricating a single-storey Kresling tube using cardboard. What has not been stressed much, if not even recognized, in the literature is the fact that such a mechanical feature is not depending only on the (chiral) geometrical pattern and unaffected by the in-plane/bending stiffness of facets and the creases’ resistance to folding. Assuming to neglect the bending stiffness of facets, in the present contribution we prove, through some numerical simulations based on a discrete model taking into account inertial terms, that only when the in-plane-to-folding stiffness ratio is large the Kresling tube exhibits the aforementioned exotic feature as described in the literature. We also prove that a low in-plane-to-folding stiffness ratio reveals: (i) an unconventional buckling mode, both for axial shortening and lengthening, which resembles the mechanism of a camera diaphragm; (ii) a kind of auxetic behavior, i.e. a stenosis in a shortening test.