The Stern-Gerlach \cite{Stern-Gerlach} experiment is notoriously counter-intuitive. The official explanation has it that the spin remains always aligned with the magnetic field such that the directions of space would be quantized: A fermion can only have its spin up or down. But that theory is based on several blatant mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is all but beyond human intuition. It is based on an understanding of spinors in SU(2) which as explained in \cite{spinors} is only Euclidean geometry. Contrary to what Pauli \cite{Pauli} has been reading into the Stern-Gerlach experiment, the directions of space are not quantized. The new and corrected paradigm, which solves all the conceptual problems, is that the fermions precess around the magnetic-field lines just like Einstein had conjectured. But surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin down as has been claimed. Indeed, despite the presence of the many different possible angles $\theta$ between the spin axis ${\mathbf{s}}$ and the magnetic field ${\mathbf{B}}$, the fermions can only have two possible energies $m_{0}c^{2}\pm\mu B$. The values $\pm\mu B$ are at variance with the continuum of values $-{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ Einstein had anticipated. What is wrong in what Einstein had expected is that the energy term $V= -{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic energy states $\pm\mu B$, and as such not valid for individual fermions. The two fermion states $\pm\mu B$ are not potential-energy states, but they are stable, just like a precessing spinning top without friction in a gravitational field is stable. We also spell out the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms.