  The mirror map in the D=3, N=4 supersymmetry connects the left and right
SU(2) automorphism groups and also the superfield representations of the
corresponding N=4 supermultiplets. The mirror N=4 harmonic superspaces use the
harmonics of two SU(2) groups and two types of the Grassmann analyticity. The
irreducible left and right N=4 supermultiplets are defined in these harmonic
superspaces. We analyze the N=4 superconformal interactions of the gauge and
matter superfields and the spontaneous breakdown of the superconformal
symmetry. The most interesting superconformal action possesses the mirror
symmetry and contains two nonlinear terms of the abelian left and right gauge
superfields, and also the mixing N=4 BF interaction which yields the
topological masses of the gauge fields and the nontrivial interaction of the
scalar and pseudoscalar fields. The minimal interactions of the left and right
N=4 hypermultiplets can be included to this abelian gauge theory. We consider
also the nonlinear N=4 gauge superfield interactions.
