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Upon straightforward four-dimensional anisotropic extension of the traditional special-relativistic two-dimensional transformations, assuming tentatively the existence of four independent, geometrically distinguished directions, we find convenient, totally anisotropic, kinematic transformations that reveal many remarkable group and invariance properties. Such a promise is shown to ground the basic manifold with the Finslerian fourth-root metric function to measure length of relativistic four-vectors. Conversion to the framework of relativistic four-momentum is also elucidated. An interesting particular algebra for subtraction and composition of three-dimensional relative velocities arises. The correspondence principle is operative in the sense that, at small relative velocities, the transformations introduced tend approximately to ordinary Lorentz precursors. Transport synchronization of clocks remains valid.