On bundles having automorphisms reversing orientations of fibers

Question

I am reading Hatcher's book on Vector Bundles and K-Theory(freely available online). I am reading about clutching functions in the first chapter and there is a statement I intuitively grasp but I am not able to prove rigorously (in order to convince myself). In order to be concrete I introduce a capture of p.25:

I am not able to prove the statement:

These bundles all have automorphisms reversing orientations of fibers.

I think that there is something here I don't understand properly. So any clarification, explanation or help would be appreciated. Thanks!

EDIT: To localize a bit further what is alarming me: I can accept the statement but the same (non-rigorous) argument/intuition which leads me to the statement would allow me to assert that every vector bundle have automorphisms reversing orientations of fibers, which, accordingly to several questions here, is false.

Answer 1

Including a bit more context, Hatcher writes (emphasis added):

The corresponding bundles are the trivial bundle and, when $n = 1$, the Möbius bundle, or the direct sum of the Möbius bundle with a trivial bundle when $n > 1$. These bundles all have automorphisms reversing orientation of fibres.

A trivial bundle admits an orientation-reversing automorphism: Fix a global frame, map the first element to its negative, and extend by linearity.

Every line bundle admits an orientation-reversing automorphism, scalar multiplication by $-1$ in the fibres.

Finally, if $E$ admits an orientation-reversing automorphism $f$, then for an arbitrary vector bundle $F$, $f \oplus I_{F}$ is an orientation-reversing automorphism of $E \oplus F$.