What are the higher dimensional analogues of left and right handedness? The dimensions of a set of three axes can be arranged in two ways; left or right handed.  Cartesian co-ordinates are by convention always oriented to comply with the right-hand rule.  It would seem this rule can be thought of as a cyclic transformation of order 3 which takes us from one axis to the next.
What are the analogues for left- and right- handedness in higher dimensions?  Particularly in infinite dimensions?
We know that three is geometrically special arising out of the parallelisability of three-sphere, and the only higher dimensional space in which this happens again is 7-dimensions so is there only an analogue in 7-dimensions or can the concept be extended to other spaces?
In particular... and this is just a bit of background perhaps not material to the question.  I'm interested in a space I'm constructing to study number theory in which every axis represents a prime number and every point along that axis represents an increment in the power of that prime number, so along the first axis we have 2, 4, 8, ... and on the 2nd axis we have 3, 9, 27, ...  By this means every co-ordinate in the infinite-dimensional space represents a unique natural number given by the product of its co-ordinates.  If the points along each axis are then spaced according to their square root, Pythagorus theorem guarantees that all points in the infinite-dimensional space are well-ordered by their distance from the origin, which is equal to the square root of the log of natural number they represent.
What I want to bring some understanding to, is how the process of counting in this space is described by some translation from any $x$ to $x+1$, and whether there might be some sense to be made of the rotation between axes that takes place with each increment.
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$"Handedness" has a natural generalization to arbitrary finite-dimensional vector spaces, or to finite-dimensional smooth manifolds, see also chirality and orientability. Martin Gardner's The New Ambidextrous Universe may be worth a look.
Let $V$ be a finite-dimensional real vector space, and $B = (\Basis_{j})_{j=1}^{n}$ an ordered basis. Given an ordered basis $B' = (\Basis_{j}')_{j=1}^{n}$, there exists a unique linear isomorphism $T:V \to V$ satisfying $T(\Basis_{j}) = \Basis_{j}'$ for each $j$. The basis $B'$ defines the same orientation as $B$ if $\det T > 0$, and defines the opposite orientation as $B$ if $\det T < 0$.
Note carefully that orientation (or handedness, or chirality) is not an intrinsic property of a single ordered basis; instead, it's a relationship between two ordered bases. Often we fix an ordered basis $B$ and declare it to be "positively-oriented", which really means that an arbitrary $B'$ is positively oriented if and only if $B'$ has the same orientation as $B$. (According to Gardner, this issue confused Kant for some time.)
This concept generalizes to finite-dimensional vector spaces over an ordered field. If your space is infinite-dimensional, you may have trouble defining the determinant of an isomorphism. If your field of scalars is not ordered, there's no way to compare $\det T$ with $0$.
In your situation, you're using the bijection from the positive integers to the set of finite sequences of non-negative integers given by the uniqueness of prime factorization: Letting $p_{j}$ denote the $j$th prime number, and letting $(k_{j})_{j=1}^{\infty}$ denote a sequence of non-negative integers with at most finitely many non-zero, you have
$$
N \sim (k_{j})_{j=1}^{\infty} \quad\text{iff}\quad
N = \prod_{j=1}^{\infty} p_{j}^{k_{j}}.
\tag{1}
$$
You can view each such sequence as an element of $\Reals^{\infty}$, the vector space of finite sequences of reals.
Your idea of considering the mapping $N \mapsto N + 1$ and the induced action on sequences of exponents in (1) doesn't appear to induce an isomorphism of the ambient vector space $\Reals^{\infty}$, however, so it's not clear whether the preceding definition of orientation will be useful to you.
