The practical way to define left and right is through weak decays of nuclei. If you take a nucleus which is unstable to $\beta$ decay, and you use a magnetic field to align the spin of the nucleus along the z-axis, the direction of charge flow in the coil generating the magnetic field defines a current loop. If you curl the fingers of your right hand along the direction of the current, the thumb points in the conventional direction of the magnetic field.
The electrons emitted during $\beta$ decay are emitted asymmetrically in the direction of the B axis. More electrons go one way than the other. The direction in which the electrons are preferentially emitted in each decay defines left and right in each $\beta$ decay experiment.
The reason for the asymmetry is ultimately because the neutrino only has one helicity, it only spins in a certain way around its direction of motion. So if you emit a neutrino/anti-neutrino in the down direction, you must lose/gain half a unit of spin in the x-y plane. Aligning the spin of the nucleus in a certain direction makes one direction easier than the other for neutrino emission.
This classic experiment was first performed by C.S. Wu in the late 1950s, after Lee and Yang suggested that parity is violated in the weak interaction. Sudarshan and Marshak, followed by Feynman and Gell-Mann, were the first to explain the phenomena by noting that the neutrino only has one helicity.
Right hand rule for EM is fake
The EM interaction respect parity (as do the strong interactions). But the description of EM elementary students get breaks parity, because it uses the right hand rule to define the magnetic field. The magnetic field does not break left-right symmetry, because you always use the right hand rule twice. Once to figure out the direction of the B field, and once to figure out the direction of the force from the B field.
It is possible to formulate EM without breaking the right hand rule, but at the cost of making B into an antisymmetric rank-2 tensor instead of an intuitive vector. So if a B field is pointing in the z-direction, the rank-2 tensor curls in the x-y plane.
But the chiral description of EM is actually understood to be more fundamental today, because of the magnetic monopoles required by quantum gravity. Magnetic monopoles break chiral symmetry for real. But we have no monopoles at low energy, so the undergraduate EM
is not formulated with all the symmetry explicit.