Positive oriented atlases characterize orientations except for one-manifolds with boundary

Suppose an orientation is selected and either $n\ge 2$, or $n>0$ and $\mathrm{\partial }M=\mathrm{\varnothing }$. By [L1] and [L2], each chart frame has a sign relative to that ray; continuity makes this sign locally constant, so restrict to its sign components. On a negative interior chart, reverse one coordinate. On a negative boundary chart with $n\ge 2$, reverse one of the first $n-1$ coordinates; this preserves ${\mathbb{H}}^n$ while reversing the frame orientation. The resulting positive charts cover $M$.

On overlaps, [L2] says that both coordinate frames are positive precisely when their change determinant is positive. Thus step 1.1 gives a positive-transition atlas. Conversely, in every $n>0$, positive transition determinants make the chart-frame rays agree on overlaps and hence define the orientation of [L1].

The converse in step 2.1 is not reversible for an arbitrary orientation when $n=1$ and there is boundary: by [L3], a positive boundary chart necessarily declares the inward vector positive. On the standard oriented interval $[0,1]$, $+{\mathrm{\partial }}_x$ is inward at $0$ but outward at $1$, so its orientation cannot be represented by positive boundary charts at both endpoints. Finally, when $n=0$, [L1] shows why independent pointwise signs, rather than the unique empty chart frame, remain the correct datum.