Magnetic confinement fusion devices, such as tokamaks and stellarators, are designed such that magnetic field lines lie in magnetic surfaces that form a foliation of nested tori. The Poincaré–Hopf index theorem implies that only surfaces with Euler characteristic χ=0 allow for vector fields that lie tangent to the surface and are smooth and nowhere-vanishing. It is often implicitly assumed that magnetic surfaces are always toroidal. In this paper, we show that surfaces with the topology of a Klein bottle (the only other compact χ=0 surface without boundary) can also occur in fusion reactor fields. These surfaces are spanned by fields that asymptotically approach and depart from closed field lines around which the mapping is reflection-hyperbolic. Configurations in which this occurs appear in the stellarator database QUASR and have been described in literature in the context of abnormal sawtooth crashes.