__The complete (__

`n`+1)-graph in`n`-dimensional spaceWe can embed the complete 2-graph with labelled vertices in a 1-dimensional world

or

and if the two "ends" of that 1-dimensional world are distinct, the above 2 embeddings are different.

We can embed the complete 3-graph with labelled vertices in a 2-dimensional world

or

and if the two "sides" of the plane are distinct, so are the 2 embeddings.

Similarly —and this will be our last example— the completed 4-graph can be embedded in 2 ways in directed 3-space —i.e. a 3-dimensional world that is distinct from its mirror image—

The first question to be raised and answered is: how do we distinguish, for each number of dimensions, between the 2 embeddings? Moreover we would like to do so in a manner that does not destroy the symmetry between the vertices.

Here the theory of inversions —i.e. (the number of) pairs of elements out of order— gives the answer. For any `k`, `k` ≥ 2, the `k`! permutation of `k` elements can be partitioned into 2 classes of equal size such that a single swap transforms any permutation of the one class into a permutation of the other class.
For elements `A`, `B`, the classes are {`AB`} and {`BA`},
for elements `A`, `B`, `C`, the classes are {`ABC`, `BCA`, `CAB`} and {`ACB`, `BAC`, `CBA`},
for elements `A`, `B`, `C`, `D`, the one class is {`ABCD`, `ACDB`, `ADBC`, `BADC`, `BCAD`, `BDCA`, `CABD`, `CBDA`, `CDAB`, `DACB`, `DBAC`, `DCBA`} and the construction of the other class is left to the reader.
I would like to stress that the existence and "shape" of these classes have nothing to do with the labels `A`,`B`,`C`,… or their alphabetic order: they are intrinsically generated by the process of permuting.
The two classes provide the answer to our first question: for any `k`, we associate the one class with the one embedding, and the other class with the other embedding.

The "oriented" (`n`+1)-graph has `n`+1 "faces", one opposite to each vertex. The way to give those faces an orientation in a systematic manner is, for instance, as follows: choose from the representative class for the (`n`+1)-graph a permutation that starts with the opposite vertex and select the rest of that permutation.
So the oriented 3-graph with class {`ABC`, `BCA`, `CAB`} has face `BC` opposite to `A`, face `CA` opposite to `B` and face `AB` opposite to `C`;
the 4-graph corresponding to `ABCD` has the 4 oriented faces corresponding to `BCD`, `ADC`, `ABD`, and `ACB` respectively, and any 2 of them contain the subface they share in opposite orientation:
e.g. `BCD` and `ADC` contain `CD` and `DC` respectively, the one having been obtained by truncating a leading `AB` (from `ABCD`, as a matter of fact), the other by truncating a leading `BA` (from `BADC`).
Generalization to more dimensions is left to the reader.