  The gauge covariant magnetic Weyl calculus has been introduced and studied in
previous works. We prove criteria in terms of commutators for operators to be
magnetic pseudo-differential operators of suitable symbol classes. The approach
is completely intrinsic; neither the statements nor the proofs depend on a
choice of a vector potential. We apply this criteria to inversion problems,
functional calculus, affiliation results and to the study of the evolution
group generated by a magnetic pseudo-differential operator.
