$w_1(\hat T X) : X \stackrel{\hat T X}{\to} B O \stackrel{w_1}{\to} B \mathbb{Z}_2
\,.$

One may identify this with the bundle that over each neighbourhood$x \in U \subset X$ of a point $x$ has as fibers the two different choices of volume forms up to positive rescaling (the two different choices of orientation).

More generally, for $E \to X$ any orthogonal group-principal bundle classified by a morphism $E : X \to \mathbf{B} O$, the corresponding orientation double cover is the $\mathbb{Z}_2$-bundle classified by

$w_1(E) : X \stackrel{E}{\to} \mathbf{B} O \stackrel{w_1}{\to} \mathbf{B} \mathbb{Z}_2
\,.$