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Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point.