Given a smooth projective curve C over an arbitrary field k, and a Brauer class $\alpha$ on C, one can construct the moduli space of $\alpha$-twisted vector bundles (of fixed rank and determinant) over C. I will discuss how given such a moduli space, one can recover the curve C and part of the Brauer class $\alpha$. Along the way, we give an explicit description of the moduli space of rank 2 twisted vector bundles with (fixed) odd determinant, on hyperelliptic curves, generalizing classical results of Desale-Ramanan. Consequently we obtain period-index bounds for certain Brauer classes on products of hyperelliptic curves,  providing evidence towards a conjecture of Colliot-Thélène. Based on joint work (in progress) with Ting Gong, Max Lieblich and Arnab Roy.