  The center-flavor symmetry of a gauge theory specifies the global form of
consistent gauge and flavor bundle background field configurations. For 6d
gauge theories which arise from a tensor branch deformation of a superconformal
field theory (SCFT), we determine the global structure of such background field
configurations, including possible continuous Abelian symmetry and R-symmetry
bundles. Proceeding to the conformal fixed point, this provides a prescription
for reading off the global form of the continuous factors of the zero-form
symmetry, including possible non-trivial mixing between flavor and R-symmetry.
As an application, we show that this global structure leads to a large class of
4d $\mathcal{N} = 2$ SCFTs obtained by compactifying on a $T^2$ in the presence
of a topologically non-trivial flat flavor bundle characterized by a 't Hooft
magnetic flux. The resulting "Stiefel--Whitney twisted" compactifications
realize several new infinite families of 4d $\mathcal{N} = 2$ SCFTs, and also
furnish a 6d origin for a number of recently discovered rank one and two 4d
$\mathcal{N} = 2$ SCFTs.
