  The distinction between the charged particle and antiparticle differential
cross sections of high-energy bremsstrahlung in the electric field of a heavy
atom is investigated. The consideration is based on the quasiclassical
approximation to the wave functions in the external field. The charge asymmetry
(the ratio of the antisymmetric and symmetric parts of the differential cross
section) arises due to the account for the first quasiclassical correction to
the differential cross section. All evaluations are performed with the exact
account of the atomic field. We consider in detail the charge asymmetry for
electrons and muons. For electrons, the nuclear size effect is not important
while for muons this effect should be taken into account. For the longitudinal
polarization of the initial charged particle, the account for the first
quasiclassical correction to the differential cross section leads to the
asymmetry in the cross section with respect to the replacement
$\varphi\rightarrow-\varphi$, where $\varphi$ is the azimuth angle between the
photon momentum and the momentum of the final charged particle.
