  In quadratic gravity, the junction conditions are six and permit the
appearance of double layer thin shells. Double layers arise typically in
theories with dipoles, i.e., two opposite charges, such as electromagnetic
theories, and appear exceptionally in gravitational theories, which are
theories with a single charge. We explore this property of the existence of
double layers in quadratic gravity to find and study traversable wormholes in
which the two domains of the wormhole interior region, where the throat is
located, are matched to two vacuum domains of the exterior region via the use
of two double layer thin shells. The quadratic gravity we use is essentially
given by a $R+\alpha R^2$ Lagrangian, where $R$ is the Ricci scalar of the
spacetime and $\alpha$ is a coupling constant, plus a matter Lagrangian. The
null energy condition, or NEC for short, is tested for the whole wormhole
spacetime. The analysis shows that the NEC is satisfied for the stress-energy
tensor of the matter in the whole wormhole interior region, notably at the
throat, and is satisfied for some of the stress-energy tensor components of the
matter at the double layer thin shell, but is not satisfied for some other
components, namely, the double layer stress-energy distribution component, at
the thin shell. This seems to mean that the NEC is basically impossible, or at
least very hard, to be satisfied when double layer thin shells are present.
Single layer thin shells are also admitted within the theory, and we present
thin shell traversable wormholes, i.e., wormholes without interior, with a
single layer thin shell at the throat for which the corresponding stress-energy
tensor satisfies the NEC, that are asymmetric, i.e., with two different vacuum
domains of the exterior region joined at the wormhole throat.
