  We extend the flyping theorem to alternating links in thickened surfaces and
alternating virtual links. The proof of the former result adapts the author's
recent geometric proof of Tait's 1898 flyping conjecture (first proved in 1993
by Menasco-Thistlethwaite) by using Boden-Karimi's recent extension of definite
surfaces (which characterize alternating classical links a la Greene) to links
in thickened surfaces. In the process, we also extend a classical result of
Gordon-Litherland, establishing an isomorphism between their pairing on a
spanning surface and the intersection form on a 4-manifold constructed as a
double-branched cover using that surface.
  To translate the flyping theorem to alternating virtual links, we establish a
new correspondence between abstract link $\textit{diagrams}$, cellularly
embedded link $\textit{diagrams}$ on closed surfaces, and equivalence classes
of virtual link $\textit{diagrams}$. We also rely on results from two companion
papers.
