Monofilar, bifilar, trifilar and quadrifilar Archimedean spiral metamaterial particles are analyzed by using point group symmetry and the methods of crystallography. From the symmetry properties electromagnetic response is determined. Magnetic, electric and magnetoelectric modes of the particles are identified along with their isotropy characteristics. Theoretical methods show that all the particles, except monofilar spiral, are non-bianisotropic. Numerical simulation results are presented to verify the analysis. Finally, effective medium theory is applied to extract the effective permeability of the spiral medium. The results indicate negative values for permeability in certain frequency ranges.