The unidirectional drift of bistable fronts (BFs) that separate two stable uniform states of a bistable system under the action of an asymmetrically oscillating zero-mean force (driver) is considered within the "pseudolinear" (piecewise-linear) model of the system. The particular case of the symmetrical (symmetrically shaped) rate functions is studied. To perform a rigorous analytic treatment for arbitrary strengths of the driving force we assume that the applied ac force is quasistatically slow. Both cases of the initially static and the initially propagating BFs are examined; various types of the "unforced" dc motion are found. We show that the unforced transport of BF takes place in any case of the asymmetric driver, whether Maxwellian construction of the rate function was balanced or not. In particular, progressive (accelerated) dc drift of the initially static BFs occurred. In contrast, both progressive and regressive (decelerated) types of unforced dc drift of the initially propagating BFs take place. Moreover, reversal of the directed motion of the initially propagating BF occurred, if the deviation of Maxwellian construction from the strictly balanced situation was relatively small; by tuning the strength of the driving force the dc drift of BF exhibits the reversal. The symmetry properties of the biharmonic driver are discussed. The biharmonic ac force consisting of a superposition of the fundamental mode and its even (odd) superharmonics is an asymmetrically (symmetrically) oscillating one. The reversal type of the unforced dc drift occurred only in the case of the even superharmonic "mixing," when the superharmonic mode of the biharmonic driver was even.