The pair $(U,\phi)$ as in the definition of orientable defines an orientation on an embedded m-dimensional manifold with boundary $M$.
Two pairs $(U,\phi)$ and $(V,\psi)$ determine the same orientation if for every local parametrization $f:W\to M$ where $W \subset H^m$ is open in $H^m$, there exists a positive function $g:W\to\mathbb R$ such that $f^(\phi) = gf^\phi$.
An orientation on $M$ is an equivalence class if pairs that determine the same orientation on $M$.
Wikidata ID: Q2748415