quantum mechanics - Why, for a spin-½ particle, are the possible outcomes of measuring spin projection along any direction the same?

If one measures the projection of spin of a spin half particle along the $x$ axis one will always get $\pm\tfrac12\hbar$.

Measuring it along the $y$ axis one will always get $\pm\tfrac12\hbar$.

Measuring it along the $z$ axis one will always get $\pm\tfrac12\hbar$.

Measuring it along "any" axis one will always get $\pm\tfrac12\hbar$.

Why are the possible outcomes the same in all directions?