Almost all the laws of physics exhibit left/right symmetry.
That is, they don't prefer one handedness over the other.
The Greek word for handedness is chirality.
Like all symmetries, the left/right symmetry is associated
with a conservation law, in this case conservation of parity.
++ Classical mechanics doesn't care about handedness.
++ Classical electrodynamics doesn't care about handedness.
++ Quantum electrodynamics doesn't care about handedness.
++ Gravitation doesn't care about handedness.
++ The strong nuclear interaction doesn't care about handedness.
-- But the weak nuclear interaction _does_ care about handedness.
This has many consequences:
If you are tempted to predict that plane-polarized light
(propagating in the Z direction) will make a spinless
test-charge move in circles in the XY plane, you'd
better think twice!
If you think Maxwell's equations require a Right-Hand
rule, you'd better think twice! I'll bet you could use
a Left-Hand rule equally well (provided you use it
consistently; don't mix & match).
If you really have discovered a handedness to the laws
of physics (other than weak nuclear interactions) you
will probably win an all-expense-paid trip to Stockholm.
All too often, we find that the real physics has a symmetry,
but some way of _writing_ the physical law does not manifest
that symmetry. The symmetry is still there, it is just not
manifest. Consider the following chain of examples:
There is nothing terribly wrong with that. It gives the
right answer, at least for rotation in a horizontal plane.
The right answer is rotationally invariant, even though the
equation _as written_ is not rotationally invariant, depending
as it does on somebody's definition of north and east.
2) Angular momentum = velocity cross position
This is better, because the rotation invariance is now
manifest. The law is written without mentioning anybody's
coordinate system.
However, the physics has a left/right symmetry that is
not manifest. You can't draw the cross product without
invoking a Right-Hand rule. But I emphasize that the
physics is still left/right symmetric!!!! The law is
just written in a misleading form.
3) Angular momentum = velocity wedge position
This is now manifestly left/right symmetric. You
can draw the wedge product in the plane, without
referring to anybody's notion of left versus right.
In fact, this definition works just fine in D=2
flatland, for which no left/right distinction can
possibly be defined. (To see this, imagine looking from
above at the flatland silhouette of a left-handed glove:
if you look at it from below it becomes right-handed.
Since in a truly flat flatland you can't define the
concepts of "above" and "below", you can't define any
reliable notion of left-handed versus right-handed.)
Of course the wedge product is manifestly rotationally
invariant, so we're OK in that department, too.
======
To summarize:
++ writing a cross product or wedge product is better
than writing out the components, because the product
is manifestly rotationally invariant, whereas the
components are not.
-- the cross product doesn't exist in D=2
-- the cross product conceals the left/right invariance
of the real physics in D=3
-- the cross product is almost never what you want
in D=1+3 spacetime.
++ the wedge product works just fine in D=2, D=3, and
D=1+3. It looks the same and means the same in all cases.
++ The wedge product allows a left/right symmetric law
to be written in a way that looks left/right symmetric.
=======================
No law of physics (still excluding weak nuclear interactions)
can make a prediction based on _one_ application of the cross
product or any other RH rule. The observable predictions always
involve _pairs_ of RH rules, so that the second one undoes the
mischief done by the first one. Example: the magnetic field
pseudovector B is defined using a cross product; this is undone
by the cross product that appears in the Lorentz force law.
The Clifford Algebra approach doesn't represent magnetism by
a pseudovector (but rather by a bivector), doesn't use cross
products in the Maxwell equations, and doesn't use cross products
in the Lorentz force law.
You can write down the axioms of Clifford Algebra without
specifying a handedness.
You have the option of _extending_ this by specifying the
handedness, e.g. by sorting the basis vectors into an ordered list,
but you don't have to (unless you are describing weak nuclear
processes); the unhanded version just treats the basis vectors
as an unordered set.
The unit pseudoscalar (i) is chiral in D=3 and higher. But I'm
claiming that almost all the laws of physics (i.e. all except
weak nuclear processes) can be written without using any such
critter. The axioms of Clifford Algebra permit but do not require
the construction of any such critter.
==================
For further reading:
The Feynman Lectures on Physics, volume III chapter 17