  The intrinsic spin Hall conductivity and the anomalous Hall conductivity of
ferromagnetic L1$_0$-CoPt are studied using first principle calculations of the
spin Berry and Berry curvatures, respectively. We find that the Berry curvature
and the spin Berry curvature exhibit different symmetry with respect to that of
the band structure. The Berry curvature preserves the $C_{4v}$ crystal rotation
symmetry along the c-axis whereas the symmetry of the spin Berry curvature
reduces to $C_{2v}$. Contributions to the Berry curvature and the spin Berry
curvature are classified by the spin character of bands crossing the Fermi
level. We find that the reduced symmetry of the spin Berry curvature is due to
band crossing points with opposite spin characters. From model Hamiltonian
analyses, we show the universality of this distinct symmetry reduction of the
spin Berry curvature with respect to the Berry curvature: it can be accounted
for based on the form of spin current operator and velocity operator in the
Kubo formula. Finally, we discuss the consequence of the reduced symmetry of
the spin Berry curvature on the relationship between the anomalous Hall and
spin Hall conductivity. When band crossing points with opposite spin characters
are present in the reciprocal space, which is often the case, the anomalous
Hall conductivity does not simply scale with the spin Hall conductivity with
the scaling factor being the spin polarization at the Fermi level.
