We report on a new mechanism involving the torsional resonance of stubs to achieve the negative effective shear modulus of an elastic metamaterial plate. Combined with a mechanism to create a negative mass density, we develop a general method to set up and enlarge a shear-horizontal-polarized double-negative branch in the elastic metamaterial plate with stubs on both sides. We explore the capabilities of this structure for polarization filtering, mode conversion and abnormal refraction. It is shown that, this metamaterial plate behaves divergently against the polarization of incident waves propagating along ΓX direction in a square lattice crystal: it behaves as a double-negative system for zero-order shear horizontal (SH0) wave but as a single-negative one for zero-order antisymmetric (A0) or symmetric (S0) Lamb waves. Mode conversion is achieved when the propagation deviates from ΓX direction. Moreover, we observe abnormal refracted patterns with both positive and negative refraction occurring at the interface between a prism-shaped supercell and the surrounding plate. Furthermore, we propose a chiral pillar to efficiently couple the torsional resonance with an incident A0 Lamb wave.