  Braiding defects in topological stabiliser codes can be used to
fault-tolerantly implement logical operations. Twists are defects corresponding
to the end-points of domain walls and are associated with symmetries of the
anyon model of the code. We consider twists in multiple copies of the 2d
surface code and identify necessary and sufficient conditions for considering
these twists as anyons: namely that they must be self-inverse and that all
charges which can be localised by the twist must be invariant under its
associated symmetry. If both of these conditions are satisfied the twist and
its set of localisable anyonic charges reproduce the behaviour of an anyonic
model belonging to a hierarchy which generalises the Ising anyons. We show that
the braiding of these twists results in either (tensor products of) the S gate
or (tensor products of) the CZ gate. We also show that for any number of copies
of the 2d surface code the application of H gates within a copy and CNOT gates
between copies is sufficient to generate all possible twists.
