It is shown that vortex core dynamics results from the interaction of two slowly deforming, but overlapping, helical vortex structures. These are the left- and right-handed components of the vortex, and are obtained by a generalized Helmholtz decomposition (the complex helical wave decomposition) of the vorticity field. This decomposition is based on a Fourier expansion in eigenfunctions of the curl operator, which has only real eigenvalues λ. The sum of eigenmodes with λ > 0 (λ < 0) constitutes the right (left) polarized component, and the vector lines of the field (e.g., vortex lines) are locally right (left) handed helixes. It is found that for a localized vortex the polarized structures are also localized, a crucial result for physical space applications. The polarized vortex structures deform slowly (compared to unpolarized structures) and behave almost like solitary waves when isolated. It is shown that this is because the nonlinearity in the Navier-Stokes equations is largely suppressed between eigenmodes of the same polarity. Moreover, the helicity of polarized structures is very high. The interaction between overlapping polarized structures however gives each structure a different propagation velocity and also results in some additional deformation. The latter is shown to occur mainly in two places: at the front of the structure where a low enstrophy bubble forms (which is a permanent feature in each of the polarized packets), and at the back where a tail develops. Otherwise, the deformation occurs on a much slower time scale compared to that for unpolarized vortices. Thus the rapid changes in the total vorticity field result from the superposition of two slowly deforming structures moving in opposite directions, as is the case for the one-dimensional (1-D) wave equation. The decomposition can also be applied to turbulent flows. In fact, it offers new insight into the structure of turbulent shear flows. The organization of small-scale vortical threads forming in the neighborhood of a coherent structure and their polarization serve as a prime example.