  The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$,
has a non-trivial kernel due to the symmetry of the manifold and presence of
trapped geodesics. For tensor fields gauge freedom increases the kernel
further, and the X-ray transform is not solenoidally injective unless $n=0$. We
characterize the kernel of the geodesic X-ray transform for $L^2$-regular
$m$-tensors for any $m\geq0$. The characterization extends to more general
manifolds, twisted slabs, including the M\"obius strip as the simplest example.
