In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? At school I learnt the Right-hand rule to remember the resulting direction of different phenomena, such as geometrical cross products, mechanical torque, or the direction a screw will move when turning it.
This directions are however decided by humans, and we could have easily defined the directions in the opposite way.
However the right hand rule also predicts phenomena which is not arbitrarily set by humans, but is intrinsic to the nature. If you have a coil in which you put some current, and the nature creates a magnetic field in a direction perpendicular to it, the direction depends on the right hand rule.
My question is, why this direction and not the opposite? Nature tends to be symmetric and not have apparent preferences for two equal options, so in a so why is one direction preferred over the other one? Why doesn't a "left hand" rule apply?
 A: The gravitational, strong nuclear, and electromagnetic forces all have parity symmetry (or, more generally, CPT symmetry), while the weak nuclear force does not. The parity symmetry of the electromagnetic force is obscured by the fact that most people, except theoretical physicists, use notation that obscures the symmetry and involves a hidden choice of handedness. This notation, which is the one you're assuming in the question, is one in which we have an electric field vector $\textbf{E}$ and a magnetic field pseudovector $\textbf{B}$. An example of notation that is manifestly parity-invariant is one in which we notate the relativistic four-force on a charge $q$ as $qFv$, where $F$ is an antisymmetric 4x4 matrix, the electromagnetic field tensor.
If the first person to set the convention (maybe Lorentz?) defining the magnetic field had made a different choice, then we would have a magnetic field that was the opposite of the one we actually have. However, the electromagnetic field tensor would be the same (assuming that the convention for the sign of charges was also the same).
For more on the mapping of tensors to vectors, see this question.
A: Classical electromagnetism is perfectly parity-invariant; it is not intrinsically left-handed or right-handed. It is true that you need to use the right hand rule to find the magnetic field, but you need to use it again to find the magnetic force. Since you always use it twice to get any directly observable quantity, the minus sign you would pick up from using the left hand rule cancels.
If we phrase everything in terms of forces, then electrostatics and magnetostatics reduce to the facts that (1) like charges repel and (2) parallel currents attract. This is clearly independent of any handedness convention. (Incidentally, the relative sign here comes from the relative sign between time and space in relativity.)

There's some deeper mathematics lurking under the surface here. Recall that the cross product of two vectors is defined as the vector pointing perpendicular to the parallelogram formed by the two vectors, with the same length as the area of the parallelogram. In three dimensions, there are two directions perpendicular to every parallelogram, which is why we need the right hand rule to pick one. In higher dimensions, this definition doesn't work at all, because there are infinitely many directions perpendicular to every plane. 
Hence the magnetic field in general dimensions can't be thought of as a vector. Instead it's better to just say it is the parallelogram itself -- it is a plane and area at every point, rather than a direction and length like a vector. The magnetic force just causes particles to rotate in the plane of the field. A current $\mathbf{J}$ at the origin creates a magnetic field at $\mathbf{r}$ in the plane spanned by $\mathbf{r}$ and $\mathbf{J}$.
Formally, these area elements are called rank $2$ differential forms. They're too involved for a beginning course, which is why we instead use the right-hand rule to convert the area to a vector, introducing an arbitrary choice. But all of the physics can be written in a manifestly symmetric way, because the phenomena really are symmetric.
A: 
However the right hand rule also predicts phenomena which is not arbitrarily set by humans, but is intrinsic to the nature, ... Basically you have a coil in which you put some current, and the nature creates a magnetic field in a direction perpendicular to it (or the equivalent experiment where you move a magnet inside a coil and induces current).

To get the full picture one has to remember that this right hand rule gives the right result for electrons and for anti-protons. Positrons and protons have to be described by a left hand rule.

My question is, why this direction and not the opposite? Nature tends to be symmetric and not have apparent preferences for two equal options, so in a so why is one direction preferred over the other one? Why doesn't a "left hand" rule apply?

The direction on which a subatomic particle gets deflected under the influence of an external magnetic field has to do with the magnetic dipole moment of these particles. Every of the described above particles has the intrinsic property of magnetic dipole moment and this moment is described as parallel or antiparallel to its spin.
So nature is not symmetric in all cases and if the electron would have the tendency to be deflected in both direction perpendicular to both the external magnetic field and the trajectory of its movement we would not be able to drive electric devices nor get an electric current from generators.
The mechanisms behind the deflection has to do with the emission of electromagnetic radiation during the deflection. You has to remember that the external magnetic field does not weaken during electron passage (any permanent magnet would work for years without loosing its strength). Imagine a bar magnet, moving into a magnetic field. The bar magnet gets aligned with this external field and so does an electron. But the electron during this alignment emits photons, all of them in the same direction and the moment of this radiation makes the electron slower and deflects it from the previous trajectory. The electron moves not in a circle, it moves in a spiral path and more than this, this path is made of tangerine slices. A positron will do the same but in the opposite direction. 
