According to Remark 2.14 in these notes, there is a self-map $s : BO_\infty \rightarrow BO_\infty$ which "exchanges the stable normal and tangential structures".
As I understand it, this means for $X$ a smooth manifold, with tangent bundle classifying map $\tau : X \rightarrow BO_{\mathrm{dim}X} \hookrightarrow BO_\infty$, and with stable normal bundle classifying map $\nu : X \rightarrow BO_\infty$, then $\tau = s \circ \nu$ and $\nu = s\circ \tau$ (only up to homotopy?).
(Here the stable normal bundle is the normal bundle to $X$ w.r.t. a smooth embedding of $X$ in some $S^m$, for $m$ large enough that all embeddings of $X$ in $S^m$ are isotopic, modulo stable equivalence; the homotopy class of the resulting $X \rightarrow BO_\infty$ should be independent of the choice of $m$ and the embedding. As described in the notes linked above, such an $m$ depending only on $\mathrm{dim}X$ can be chosen following a theorem of Whitney.)
Would anyone be able to suggest a hint on how to find such a map $s$? Or, could anyone suggest a source that explains the details of this construction?
It is simpler than you might expect. $BO_\infty$ classifies stable vector bundles. As we know, the monoid of stable vector bundles over a space $X$ is an abelian group. We can then define a natural transformation from the stable vector bundle functor to itself which on morphisms is the identity and which on objects sends a stable vector bundle to its inverse.
The Yoneda lemma implies that there is a corresponding map $BO_\infty \rightarrow BO_\infty$ representing this natural transformation.