I've been working my way through Leon Simon's geometric measure theory notes and started wondering in how far (or, rather, why) the regular part $\operatorname{reg} T$ of an $n$-current $T \in D_n(\mathbb{R}^N)$ which is area-minimizing in a (possibly orientable) submanifold $M^m \subset \mathbb{R}^N$ is necessarily oriented when considered as an embedded manifold. (Possibly this holds only when $T$ has codimension 1 in $M$, i.e. when $m = n+1$?)
The reason I think this is true is that on the first page of appendix B, Leon Simon states that an area-minimizing cone $C$ of codimension 1 in $\mathbb{R}^{n+1}$ is necessarily orientable on its regular set $\operatorname{reg} C$. Moreover, I've found this web page where the author says that
Non-orientable complete minimal surfaces are never globally embedded.
But a regular area-minimizing 2-current in $\mathbb{R}³$ defines a complete, embedded (and even stable) minimal surface (without boundary), so if the author's statement is true, it must also be orientable.
At the same time, I don't see any obstruction that would prevent, say, the Möbius strip from defining a regular current, since the $n$-vector valued orientation function $\vec{T}$ of a current $T$ (compare eq. 2.7 of chapter 6 of Simon's notes) only needs to be $ℋ^n$-measurable, so picking a non-continuous "orientation" $\vec{T}$ should not pose a problem.
Hence, I conclude that it must be the property of being area-minimizing that enforces global orientability on the regular set. But how exactly does it do that?
The regular set of a $k$-current $T$ is always oriented. Locally, in a neighborhood $T$ of a point $x ∈ \operatorname{reg}(T)$, this is clear as the restriction $T|_V$ can be written as $T|_V = m ⟦N⟧$ for some oriented $k$-dimensional submanifold $N$ embedded in $V$, so the orientation function $\vec{T}$ can be taken to be smooth there. Now if these oriented pieces of $\operatorname{reg}(T)$ did not clue together to a globally oriented embedded submanifold, $\vec{T}$ would make a discontinuous jump somewhere in $\operatorname{reg}(T)$. But then this causes the boundary $∂T$ to have non-zero support where the jump occurs (use Stokes' theorem) and since the boundary is always part of the singular set but the jump occurred on the regular set, this results in a contradiction.
In particular, taking the Möbius strip as a current, this current will never be regular everywhere.