A fixed homeomorphism admits groupoid compatibility exactly when both directions are differentiable

Let $\mathbb k$ be any nontrivially normed field, let $E$ be a normed additive commutative group with a normed $\mathbb k$-vector-space structure, let $H$ be a topological space, and let $I$ be a model with corners from $H$ to $E$: its forward map is a continuous embedding defined on all of $H$, its inverse is a continuous total map $E\to H$ inverse on the range, and its range has nonempty interior; that range is convex over the induced real scalars when $\mathbb k$ has a compatible real- or complex-field structure, and is all of $E$ otherwise. Let $n$ be any nonnegative integer, $\infty$, or $\omega$, denoting respectively finite differentiability order, smoothness, or analyticity, and let $M,M'$ be topological spaces with chosen atlases and preferred local charts into $H$, both satisfying the $C^n$ manifold compatibility condition for $I$. Here that compatibility means every atlas transition $\psi$ and its inverse become $C^n$ in $E$ after conjugation by $I$, on $I(\operatorname{source}\psi)$ and $I(\operatorname{target}\psi)$ respectively. For every homeomorphism $e:M\to M'$, there exists a homeomorphism $h:M\to M'$ equipped with the property that every partial coordinate expression $c'\circ h\circ c^{-1}$, for atlas charts $c,c'$ on $M,M'$, and its inverse become $C^n$ after conjugation by $I$ on their respective model-coordinate domains, and whose underlying homeomorphism is exactly $e$, if and only if both $e$ and $e^{-1}$ are $C^n$ manifold maps relative to $I$. For a map $f$ in either direction, the latter condition means that, at every point $x$ in its domain, $f$ is continuous at $x$ and $\kappa_{f(x)}\circ f\circ\kappa_x^{-1}$ is $C^n$ at $\kappa_x(x)$ within $I(H)$, where each $\kappa$ is the corresponding preferred chart followed by $I$. The existential assertion requires existence, with no uniqueness assertion. It includes $n=0$, empty coordinate overlaps, and empty spaces; an empty space forces both spaces to be empty because $e$ is a homeomorphism. Neither finite-dimensionality nor completeness of $E$ is assumed.