We consider planar σ-harmonic mappings, that is mappings U whose components
u<su>p<small>1</small></sup> and u<sup><small>2</small></sup> solve a divergence structure 
elliptic equation div(σ&nabla;u<sup><small>i</small></sup>)=0, for i = 1, 2.
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We investigate whether a locally invertible σ-harmonic mapping U is also quasiconformal.
Under mild regularity assumptions, only involving det σ and the antisymmetric part of σ,
we prove quantitative bounds which imply quasiconformality