Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then we obtain a stable normal framing of the normal bundle $\nu(j)$ by

$$ \varepsilon^m \oplus \nu(j) \cong TM\oplus \nu(j) \cong \varepsilon^n|_M. $$

Any other embedding $j'\colon M\to \mathbb R^{n'}$ is isotopic to $j$ if we consider $j$ and $j'$ as maps into some $\mathbb R^l$ for $l$ big enough. The isotopy provides a cobordism $W$ between $j(M)$ and $j'(M)$ which is parallelizable and hence it has a stable normal framing which restricts to the stable normal framings of $\nu(j)$ and $\nu(j')$ at the boundary of $W$. Thus $M$ together with the chosen trivilization represents a well-defined element in the bordism group of stably framed $m$-dimensional manifolds $\Omega_m^{\text{fr}}$.

**Question:** We have $\Omega_1^{\text{fr}} \cong \pi_1^S\cong \mathbb Z_2$. Do the two classes of $\Omega_1^{\text{fr}}$ depend on the orientation of the circle? If no, how do I distinguish these two classes by simply framing the tangent bundle of a circle?