The random walk of turbulent magnetic field lines strongly affects transport of energetic particles in astrophysical plasmas, but is not well understood for general configurations that lack rotational symmetry. Here we derive nonperturbative field-line diffusion coefficients for magnetic fluctuations that are nonaxisymmetric with respect to the mean magnetic field. We consider a superposition of slab plus two-dimensional fluctuations, a model that has proven useful in heliospheric studies. Two independent parameters are introduced to allow polarization of the slab component and stretching of the two-dimensional component. With the assumptions of homogeneity, the diffusion approximation, and Corrsin's independence hypothesis, we derive two coupled biquadratic equations for the diffusion coefficients. The results and underlying assumptions are confirmed by numerical simulations. Special cases of interest include the counterintuitive results that enhanced fluctuations in one direction lead to decreased diffusion in the other direction, and that extreme nonaxisymmetry leads to diffusion coefficients proportional to the rms two-dimensional fluctuation.