  Recently, Duncan and Mack-Crane established an isomorphism, as Virasoro
modules at central charges c=12, between the space of states of the Conway
Moonshine Module and the space of states of a special K3 theory that was
extensively studied some time ago by Gaberdiel, Volpato and the two authors. In
the present work, we lift this result to the level of modules of the extensions
of these Virasoro algebras to N=4 super Virasoro algebras. Moreover, we relate
the super vertex operator algebra and module structure of the Conway Moonshine
Module to the operator product expansion of this special K3 theory by a
procedure we call reflection. This procedure can be applied to certain
superconformal field theories, transforming all fields to holomorphic ones. It
also allows to describe certain superconformal field theories within the
language of super vertex operator algebras. We discuss reflection and its
limitations in general, and we argue that through reflection, the Conway
Moonshine Module inherits from the K3 theory a richer structure than
anticipated so far. The comparison between the Conway Moonshine Module and the
K3 theory is considerably facilitated by exploiting the free fermion
description as well as the lattice vertex operator algebra description of both
theories. We include an explicit construction of cocycles for the relevant
charge lattices, which are half integral. The transition from the K3 theory to
the Conway Moonshine Module via reflection promotes the latter to the role of a
medium that collects the symmetries of K3 theories from distinct points of the
moduli space, thus uncovering a version of symmetry surfing in this context.
