Oriented clutching classifies oriented bundles over spheres

Each hemisphere disk is connected and the restriction of the oriented bundle is trivial by the finite argument in [F1]. If a chosen trivialization reverses orientation, compose it with one fixed reflection; hence both hemisphere trivializations may be chosen orientation-preserving. Their equatorial transition then lies in ${\mathrm{GL}}_n^{+}(\mathbb{R})$ by [F2].

Repeat the equivalence calculation of [F1] using only orientation-preserving hemisphere gauges. Such gauges take values in ${\mathrm{GL}}_n^{+}$, which is path connected, and their disk restrictions are nullhomotopic. Thus two positive clutching maps give orientation-preservingly isomorphic bundles exactly when they are homotopic, proving the first classification. This includes $k=1$: every map from $S^0$ to the path-connected group has the single unbased homotopy class.

Polar normalization preserves determinant sign and deformation retracts ${\mathrm{GL}}_n^{+}(\mathbb{R})$ onto $\mathrm{SO}(n)$. Equivalently, it orthonormalizes the positive frames in [F3]. It therefore induces the second displayed bijection on homotopy classes.

Fix a reflection $r\in {\mathrm{GL}}_n(\mathbb{R})$. After reversing the chosen orientation of every fiber, the old oriented hemisphere trivializations become orientation-reversing; postcomposing both with $r$ makes them oriented again. If their old transition was $g$, the new one is $rg{r}^{-1}$. A different reflection differs from $r$ by positive matrices and gives the same action on the classified homotopy classes. The argument is finite and uses no choice principle.