Instead we can distinguish fermions and bosons by the sort of classical field equations that they obey. Bosons obey second-order partial differential equations; for example, the Klein-Gordon equation,
or Maxwell equations, when thought of as equations for the potential:
On the other hand, fermions obey first-order equations like the Dirac (or Weyl) equations:
These equations are of course satisfied by free fields, but interactions only add lower order terms and hence do not change the order of the differential equation.
We see therefore that a symmetry which relates bosons to fermions is relating solutions of second-order equations to solutions of first-order equations.