Another way to visualize electron spin is to consider the "Dirac Electron"
Dirac's one equation for a massive particle can be rewritten as two equations for two interacting massless particles, where the coupling constant of the interaction is the mass of the electron.
We can follow these rules and visualize a current loop capturing an orthogonal magnetic field. This way you can represent both electrons and anti-electrons as left or right handed particles. These particles can also exist as either up or down state relative to its environment.
These coupled spinors spin orthogonal to themselves. From the direction of spin, you can see there is no way to push the electron and anti-electron together without destroying them.
Also note, turning the inner (blue) loop 360 degrees, only turns the larger (red) loop 180 degrees leaving it upside down. You must turn another 360 degrees (720 total) before it is right side up.