  The squares of the three components of the spin-s operators sum up to
$s(s+1)$. However, a similar relation is rarely satisfied by the set of
possible spin projections onto mutually orthogonal directions. This has
fundamental consequences if one tries to construct a hidden variable (HV)
theory describing measurements of spin projections. We propose a test of local
HV-models in which spin magnitudes are conserved. These additional constraints
imply that the corresponding inequalities are violated within quantum theory by
larger classes of correlations than in the case of standard Bell inequalities.
We conclude that in any HV-theory pertaining to measurements on a spin one can
find situations in which either HV-assignments do not represent a physical
reality of a spin vector, but rather provide a deterministic algorithm for
prediction of the measurement outcomes, or HV-assignments represent a physical
reality, but the spin cannot be considered as a vector of fixed length.
