  We apply Shubin's theory of global symbol classes $\Gamma_{\rho}^{m}$ to the
Born-Jordan pseudodifferential calculus we have previously developed. This
approach has many conceptual advantages, and makes the relationship between the
conflicting Born-Jordan and Weyl quantization methods much more limpid. We
give, in particular, precise asymptotic expansions of symbols allowing to pass
from Born-Jordan quantization to Weyl quantization, and vice-versa. In addition
we state and prove some regularity and global hypoellipticity results.
