  This work constructs a class of non-symmetric periodic Schr\"odinger
operators on metric graphs (quantum graphs) whose Fermi, or Floquet, surface is
reducible. The Floquet surface at an energy level is an algebraic set that
describes all complex wave vectors admissible by the periodic operator at the
given energy. The graphs in this study are obtained by coupling two identical
copies of a periodic quantum graph by edges to form a bilayer graph.
Reducibility of the Floquet surface for all energies ensues when the coupling
edges have potentials belonging to the same asymmetry class. The notion of
asymmetry class is defined in this article through the introduction of an
entire spectral A-function $a(\lambda)$ associated with a potential--two
potentials belong to the same asymmetry class if their A-functions are
identical. Symmetric potentials correspond to $a(\lambda)\equiv0$. If the
potentials of the connecting edges belong to different asymmetry classes, then
typically the Floquet surface is not reducible. An exception occurs when two
copies of certain bipartite graphs are coupled; the Floquet surface in this
case is always reducible. This includes AA-stacked bilayer graphene.
