  Ba\~nados $et\, al.$ (BSW) found that Kerr black holes can act as particle
accelerators with collisions at arbitrarily high center-of-mass energies.
Recently, collisions of particles with spin around some rotating black holes
have been discussed. In this paper, we study the BSW mechanism for spinning
particles by using a metric ansatz which describes a general rotating black
hole. We notice that there are two inequivalent definitions of center-of-mass
(CM) energy for spinning particles. We mainly discuss the CM energy defined in
terms of the worldline of the particle. We show that there exists an
energy-angular momentum relation $e = \Omega_h j$ that causes collisions with
arbitrarily high energy near-extremal black holes. We also provide a simple but
rigorous proof that the BSW mechanism breaks down for nonextremal black holes.
For the alternative definition of the CM energy, some authors find a new
critical spin relation that also causes the divergence of the CM mass. However,
by checking the timelike constraint, we show that particles with this critical
spin cannot reach the horizon of the black hole. Further numerical calculation
suggests that such particles cannot exist anywhere outside the horizon. Our
results are universal, independent of the underlying theories of gravity.
