Small-scale intermittency is a defining feature of fully developed fluid turbulence, marked by rare and extreme fluctuations of velocity increments and gradients that defy mean-field descriptions. Existing multifractal descriptions of intermittency focus primarily on longitudinal increments and gradients, despite mounting evidence that transverse components exhibit distinct and stronger intermittency. Here, we develop a unified multifractal framework that jointly prescribes longitudinal and transverse velocity increments, and extends to gradients, We derive explicit relations linking inertial-range scaling exponents of structure functions to moments of velocity gradients in dissipation range, which are consistent with known exact third moment relations ($4/5$-th and $4/15$-th laws) and dissipation anomaly. Our results reveal that longitudinal gradient scaling is solely prescribed by longitudinal structure functions, as traditionally expected; however, transverse gradient scaling is prescribed by mixed longitudinal-transverse structure functions. Validation with high-resolution direct numerical simulations of isotropic turbulence, at Taylor-scale Reynolds number up to 1300 demonstrates excellent agreement, paving way for a more complete and predictive description of intermittency faithful to the underlying turbulence dynamics.