You have suggested that mirror-particles of one another, i.e. particles which look like the mirror image of one another, are actually commutative if we introduce additional dimensions. For instance, a 2D spiral with right orientation can be turned upsidedown in 3D to produce a 2D spiral with left orientation (clockwise rotation of the spiral) (1). I asked whether the existence of zillions of chiral molecules, i.e. molecules which are mirror images of one another, hints that no extra spatial dimensions exist in molecular scales.
Moreover, string theory suggests there are several additional spatial dimensions which are 'comapctified' - like a closed string with a small but finite radius. Thus, chiral particles smaller than the radius of these dimensions should be able to switch to their mirror-particles and back, i.e. their wavefunction should commute with the mirroring operator.

(1) Riddle of the Day - Why is the counterclockwise direction defined as the positive mathematical direction?

I'm not deeply familiar with string theory, but I can imagine a small closed loop, let's say a circle, which looks from a distance like a dot. I can also imagine a cylinder with a finite small radius which looks from a distance like a line. That cylinder has one infinite dimension running along it, which is visible at any scale, and another compact dimension which is visible only on a scale small enough. This compact dimension must be quantized, as phi(r,z) should be equal to phi(r+2pi*n*R,z) for any r, where R is the radius of the comapct dimension.

The mirroring operator (let's tag it P) simply flips x with -x, like a mirror. which means Px=-x (that's an eigenvalue equation, -1 is the eigenvalue of P). Please warn me if I'm relating to stuff you haven't previously encountered, when I say for instance that x^2 is conserved under mirroring, thus x^2 commutes with P. Px^2=x^2P, thus [P,x^2]=0 - the commutator is 0. Any symmetric function surely commutes with P.
If a particle has an extra dimension through which to flip upsidedown to its mirror image, then it should be conserved under mirroring. Its wavefunction should commute with P.

Now, since we know chiral molecules do not spontaniously switch to one another, then I assume we can deduce they have no extra dimension to 'flip through'. If string theory allows particles to 'flip through' those compact dimensions, which is something I honestly don't know, then we can limit the radius of these dimensions to the lowest scale at which we can tell that a particle doesn't flip to its image. Spin-up electrons do flip to spin-down states, but molecules don't flip to their mirror-molecules.