Let $(M,g)$ be a $d$-dimensional, oriented pseudo-Riemannian manifold, and $V$ the subbundle of $TM\oplus T^*M$ given by the graph of the linear isomorphism $g^\sharp:TM\rightarrow T^*M$ associated to the metric $g$. We know that the bundle $TM\oplus T^*M$ carries a natural pseudo-Riemannian metric $h$ of signature $(d,d)$ and admits a spin structure associated to $h$ - for example, the (space of sections of the) exterior algebra bundle $\Lambda^*T^*M$ is a Clifford module, and its tensor product $\Lambda^*T^*M\otimes(\Lambda^dT^*M)^{1/2}$ with the real line bundle of half-densities over $M$ is a spinor bundle. Moreover, the space of spin structures on $(TM\oplus T^*M,h)$ is an affine space modelled on the group $H^1(M,\mathbb{Z}_2)$ of real line bundles over $M$, which maps the corresponding spinor bundles onto each other by tensoring (the above preliminary results can be found on Chapter 2 of Marco Gualtieri's PhD thesis on generalized complex geometry, arXiv:math.DG/0401221).

Question(s): if $(M,g)$ admits a spin structure, does a choice of spin structure on $(TM\oplus T^*M,h)$ descend by restriction to $V$ to a choice of spin structure on $(M,g)$? Does this establish a one-to-one correspondence between both sets of spin structures? If so, how does this generalize to, say, generalized Riemannian metrics (i.e. rank-$d$ subbundles $W$ of a twisting of $TM\oplus T^*M$ by a Cech 1-cocycle $B$ with values at closed 2-forms, such that the restriction of $h$ to $W$ is positive definite)?