  We define the worldline harmonic SU(2|1) superspace and its analytic subspace
as a deformation of the flat N=4, d=1 harmonic superspace. The harmonic
superfield description of the two mutually mirror off-shell (4,4,0) SU(2|1)
supermultiplets is developed and the corresponding invariant actions are
presented, as well as the relevant classical and quantum supercharges. Whereas
the \sigma-model actions exist for both types of the (4,4,0) multiplet, the
invariant Wess-Zumino term can be defined only for one of them, thus
demonstrating non-equivalence of these multiplets in the SU(2|1) case as
opposed to the flat N=4, d=1 supersymmetry. A superconformal subclass of
general SU(2|1) actions invariant under the trigonometric-type realizations of
the supergroup D(2,1;\alpha) is singled out. The superconformal Wess-Zumino
actions possess an infinite-dimensional supersymmetry forming the centerless
N=4 super Virasoro algebra. We solve a few simple instructive examples of the
SU(2|1) supersymmetric quantum mechanics of the (4,4,0) multiplets and reveal
the SU(2|1) representation contents of the corresponding sets of the quantum
states.
