It seems to be that the formal definition of a Δ-set doesn't forbid identifying different faces of a simplex, that is, face maps $d_i$ and $d_j$, $i \ne j$ may map element $a : S_n$ to a same element $b : S_{n-1}$. This means that we can have, for example, a 1-simplex $[v_0, v_0]$ that forms a loop.
That also seems to imply that the orientation isn't defined in such cases, since it is usually induced by the order on the constituent vertices.
Does this mean that there are delta complexes representable by the formal structure that don't have unique correspondence to a topological space? I'm thinking of different gluings of the dunce cap as an example where the edges are identified; the orientation of the gluing seams seems to affect the resulting topology.
The geometric realization of 1-simplex forming a loop is: (denoting $p \in Δ^0$ as the only point in the 0-simplex)
$|S|= \{ ({e_0} \times \Delta ^1), (v_0, p)\} /_{{\sim }}$
With $\sim$ defined as:
$(e_0, d^0p)\sim (v_0, p)$
$(e_0, d^1p)\sim (v_0, p)$
The points $d^0p$ and $d^1p$ of $\Delta ^1$ would have to be the same point, which means that the 1-simplex corresponding to $e_0$ would need to be of size zero. This resolves the issue.
It seems in general that faces of the same simplex that get identified in an orientation-conflicting way, would indeed collapse to points when geometrically realized. I can see the motivation for more restricted simplical complexes because of this.