  We consider two dimensional $\mathcal{N}=(4,4)$ superconformal field theories
in the moduli space of symmetric orbifolds of K3. We complete a classification
of the discrete groups of symmetries of these models, conditional to a series
of assumptions and with certain restrictions. Furthermore, we provide a partial
classification of the set of twining genera, encoding the action of a discrete
symmetry $g$ on a space of supersymmetric states in these models. These results
suggest the existence of a number of surprising identities between seemingly
different Borcherds products, representing Siegel modular forms of degree two
and level $N>1$. We also provide a critical review of various properties of the
moduli space of these superconformal field theories, including the groups of
dualities, the set of singular models and the locus of symmetric orbifold
points, and describe some puzzles related to our (lack of) understanding of
these properties.
