  In this article, we unravel an intimate relationship between two seemingly
unrelated concepts: elasticity, that defines the local relations between stress
and strain of deformable bodies, and topology that classifies their global
shape. Focusing on Moebius strips, we establish that the elastic response of
surfaces with non-orientable topology is: non-additive, non-reciprocal and
contingent on stress-history. Investigating the elastic instabilities of
non-orientable ribbons, we then challenge the very concept of
bulk-boundary-correspondence of topological phases. We establish a quantitative
connection between the modes found at the interface between inequivalent
topological insulators and solitonic bending excitations that freely propagate
through the bulk non-orientable ribbons. Beyond the specifics of mechanics, we
argue that non-orientability offers a versatile platform to tailor the response
of systems as diverse as liquid crystals, photonic and electronic matter.
