We define homogeneous classes of x-dependent anisotropic symbolṡ S m γ,δ (A) in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent

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... symbols inṠ 0 1,1 (A) yield Calderón-Zygmund kernels, yet their L 2 boundedness fails. Finally, we prove boundedness results for the classṠ m 1,1 (A) on weighted anisotropic Besov and Triebel-Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)]. 2010 Mathematics Subject Classification: Primary 47G30, 42B20, 43A85; Secondary 42B15, 42B35, 42C40. Key words and phrases: anisotropic symbol, multiplier operator, pseudodifferential operator, Calderón-Zygmund operator, space of homogeneous type.