  From atomic crystals to bird flocks, most forms of order are captured by the
concept of spontaneous symmetry breaking. This paradigm was challenged by the
discovery of topological order, in materials where the number of accessible
states is not solely determined by the number of broken symmetries, but also by
space topology. Until now however, the concept of topological order has been
linked to quantum entanglement and has therefore remained out of reach in
classical systems. Here, we show that classical systems whose global geometry
frustrates the emergence of homogeneous order realise an unanticipated form of
topological order defined by non-orientable order-parameter bundles:
non-orientable order. We validate experimentally and theoretically this concept
by designing frustrated mechanical metamaterials that spontaneously break a
discrete symmetry under homogeneous load. While conventional order leads to a
discrete ground-state degeneracy, we show that non-orientable order implies an
extensive ground-state degeneracy -- in the form of topologically protected
zero-nodes and zero-lines. Our metamaterials escape the traditional
classification of order by symmetry breaking. Considering more general stress
distributions, we leverage non-orientable order to engineer robust mechanical
memory and achieve non-Abelian mechanical responses that carry an imprint of
the braiding of local loads. We envision this principle to open the way to
designer materials that can robustly process information across multiple areas
of physics, from mechanics to photonics and magnetism.
