The differential structure of operator bases used in various forms of the Weyl-Wigner-Groenewold-Moyal (WWGM) quantization is analyzed and a derivative-based approach, alternative to the conventional integral-based one is developed. Thus the fundamental quantum relations follow in a simpler and unified manner. An explicit formula for the ordered products of the Heisenberg-Weyl algebra is obtained. The W-infinity-covariance of the WWGM-quantization in its most general form is established. It is shown that the group action of W-infinity that is realized in the classical phase space induces on bases operators in the corresponding Hilbert space a similarity transformation generated by the corresponding quantum W-infinity which provides a projective representation of the former W-infinity. Explicit expressions for the algebra generators in the classical phase space and in the Hilbert space are given. It is made manifest that this W-infinity-covariance of the WWGM-quantization is a genuine property of the operator bases. (C) 1997 American Institute of Physics.