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Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has a good summary of the preceding chapters the spin value has a relation to the representations of the Poincaré group, which encodes the spacetime symmetry that each particle observes.
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These symmetries can be characterized by small integer numbers:
  • spin 0: representation
  • spin half: representation
  • spin 1: representation
As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
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