Classes of pseudodifferential operators depending on the anisotropy vector \(\varvec{\varkappa }\) are considered. These operators continuously act on a consistent scale of anisotropic Hölder and Hölder-Zygmund spaces. In the isotropic case, when the vector \(\varvec{\varkappa =}(\varvec{1,}\varvec{1,}\varvec{\ldots ,}\varvec{1})\) , the introduced operator classes coincide with the Hörmander class \(\varvec{S}^{\varvec{m}}_{\varvec{1,} \varvec{0}}\) , and the spaces coincide with the classical Hölder-Zygmund spaces. The main result is the Beals-type characterization of anisotropic pseudodifferential operators in terms of boundedness of their commutators with differentiation operators and multiplication operators by coordinate functions. This characterization is applied to prove that the introduced classes of anisotropic pseudodifferential operators are closed under taking the inverse operator when it exists.
