Orientation reversal is the connected-sum inverse

By [L1] the punctured homotopy sphere $P=\mathrm{\Sigma }\setminus \mathrm{int}{D}^n$ is a compact contractible smooth $n$-manifold with boundary $\mathrm{\partial }P=S^{n-1}$.

Round $P\times [0,1]$ explicitly in the collars of [L2]. Near either corner, let $s,t\ge 0$ be the two inward coordinates and put $w=s-t$, $z=s+t$, so the quadrant is $z\ge \mathrm{\mid }w\mathrm{\mid }$. Choose a smooth convex function $g(w)\ge \mathrm{\mid }w\mathrm{\mid }$ equal to $\mathrm{\mid }w\mathrm{\mid }$ outside a small interval (convolve $\mathrm{\mid }w\mathrm{\mid }$ with a nonnegative even smooth kernel of integral one and small compact support). Replace $z\ge \mathrm{\mid }w\mathrm{\mid }$ by $z\ge g(w)$, using the same profile over $\mathrm{\partial }P$ and disjoint neighborhoods at the two ends. The graph is smooth and agrees with the faces away from the corner, so the retained region $V$ is a compact smooth manifold with boundary. The homotopy $z\mapsto z+u\max \{0,g(w)-z\}$, $0\le u\le 1$, extended by the identity, retracts the original product onto $V$; its support stays inside the chosen collar. Thus $V$ is contractible. Its boundary joins two shortened copies of $P$ by the boundary collar cylinder, hence is the double of $P$, namely $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$; collar reparametrizations identify the shortened copies with $P$. Choose the orientation of $V$ so the first copy has the orientation of $\mathrm{\Sigma }$; the other copy then has the opposite orientation.

Remove from $V$ the interior of a small smoothly embedded $(n+1)$-disk meeting only the interior, obtaining the compact oriented manifold $C$ whose boundary has the face $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$ and a standard sphere face $S^n$.

Using collar thickenings, excision identifies $H_{\ast }(C,S^n)$ with $H_{\ast }(V,D^{n+1})=0$ because $V$ and the disk are contractible, so the inclusion $S^n\to C$ is an integral homology isomorphism; the boundary relation of [L3] gives $[\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })]+[S^n]=0$ in $H_n(C)\cong \mathbb{Z}$, so the other face inclusion also induces an isomorphism on $H_n$ and hence on all reduced homology, since $C$ has the homology of a point in intermediate degrees.

Van Kampen applied after reattaching the removed disk shows ${\pi }_1(C)={\pi }_1(V)=0$, and both faces are simply connected: $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$ by the connected-sum lemma and $S^n$ for $n\ge 5$.

The compact smooth manifold $C$ has a finite CW model by [L1]'s finite-CW clause, so the simply connected homology Whitehead criterion upgrades both face inclusions to homotopy equivalences; hence $C$ is an h-cobordism from $S^n$ to $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$, proving that orientation reversal is the inverse.