Is it possible to represent a set of generic things coherently without implicitly creating an order on them? Sorry if this question seems a little incoherent, I'm not certain of the proper terminology here. I've seen unordered sets represented in various ways, but it recently occurred to me that most of these ways, while not presenting an explicit order, do imply or allow implicit orders on the elements of the set. For example:
(a, A, 1) has the implicit orderings of right-to-left and left-to-right
-a
-A     has the implicit orderings of top-to-bottom and bottom-to-top
-1
the spoken phrase 'a, A, 1' has the implicit orderings first-said-to-last-said and last-said-to-first-said
...and so on. The only ways I can think of to represent data without such an implicit ordering, for example saying 'a', 'A' and '1' at the same time, involve some operation equivalent to addition, which can render extraction of the data impossible or ambiguous.
There is a kind-of-exception to this. If the data is written along the edge of a circle, there is no longer any starting point for a potential ordering, though each bit of data is still implicitly associated with a 'clockwise neighbour' and 'counterclockwise neighbour'. Changing to an N-sphere where N is the number of elements in the set averts this, if elements are placed at 'poles' of the sphere and therefore are equidistant from each other, but for sets of more than three objects the problem of actually representing it grows very difficult.
Is there any such way to represent an unordered set of more than three objects in three or fewer dimensions of space?
 A: Let me try to formalize your notion of "representation of a set" in a way that makes your question answerable.  If $A$ is a set, then let us say a representation of $A$ consists of a set $X$, a group $G$ acting on $X$, and a function $f:A\to X$.  Given a representation, we write $G_A$ for the group of permutations of $A$ which are induced by elements of $g$.  That is, a permutation $\sigma:A\to A$ is in $G_A$ if there exists $g\in G$ such that $g(f(a))=f(\sigma(a))$ for all $a\in A$.  If $f$ is injective, let us say the representation is faithful.
We can think of $G_A$ as a measure of how much we can distinguish the elements of $A$ under the representation $f$.  If $G_A$ consists only of the identity, then we can distinguish all the elements of $A$ from each other completely, so (in your language) any ordering of $A$ is "implicit" in the representation.  If $G_A$ is the entire symmetric group $S_A$, then the elements of $A$ are totally indistinguishable (or totally symmetric) under the representation.  In general, the larger $G_A$ is, the more "indistinguishable" the elements of $A$ are under the representation.
The specific question you ask at the end can then be formalized as follows:

Let $X=\mathbb{R}^n$ for some $n\leq 3$ and let $G$ be the group of Euclidean motions acting on $X$.  For what sets $A$ is there a faithful representation of $A$ in $X$ such that $G_A=S_A$?  More generally, what subgroups $G_A\subseteq S_A$ can be realized by faithful representations of $A$ in $X$?

Let me answer the first question and say a bit about the second more general question.  If you have a representation of a set $A$ in $\mathbb{R}^n$ (with $G$ the group of Euclidean motions) such that $G_A=S_A$, or more generally such that $G_A$ is 2-transitive, then the Euclidean distances $d(f(a),f(b))$ for distinct $a$ and $b$ must all be equal.  If the representation is faithful, it follows that $|A|\leq n+1$ and the points $f(A)$ are the vertices of a regular $n$-simplex (for instance, you can easily prove this by induction on $n$).  So in particular, if $n\leq 3$ and the representation is faithful, then $G_A$ can only be 2-transitive if $|A|\leq 4$.  Concretely speaking, you can represent $4$ objects "indistinguishably" in 3-dimensional space by putting them at the vertices of a regular tetrahedron, but you cannot do this with more than $4$ points.
As you note, however, you can always represent any finite set faithfully in $\mathbb{R}^2$ such that $G_A$ acts transitively on $f(A)$, by putting the points evenly spaced around a circle.  In this case, $G_A$ is the dihedral group $D_{|A|}$.
In general, if $f:A\to\mathbb{R}^n$ is a representation of a finite set in Euclidean space, let $b$ be the barycenter of the points $f(A)$ and let $H$ be the linear affine space spanned by $f(A)$.  Then $H$ is a Euclidean space of dimension $k\leq n$, and every element of $G_A$ is realized by a unique isometry of $H$, and such isometries necessarily fix the point $b$.  Identifying $b$ as the origin, such isometries can be extended canonically to $\mathbb{R}^n$ by demanding that they fix $H^\perp$, so we get a representation $G_A\to O(n)$.  Furthermore, there is a canonical $G_A$-equivariant map $\mathbb{R}^{|A|}\to H$ sending the standard basis to the image of $f$.  Conversely, given such a $G_A$-equivariant map to a representation of $G_A$, we can recover $f$, and the group associated to $f$ will at least contain $G_A$.
So, in the language of representation theory, we can restate the notion of a "representation of $A$ on $\mathbb{R}^n$ such that $G_A$ contains a given subgroup $H\subseteq S_A$" as follows.  Such a representation simply consists of an $n$-dimensional real representation $V$ of the group $H$, together with a map $\mathbb{R}^{|A|}\to V$ of $H$-representations from the permutation representation of $H$ associated to the inclusion $H\to S_A$.  I don't know much about what you can say about such representations in general, but I expect this and similar ideas have been well-studied in representation theory.
