Sturm-sequence properties are reported for recurrence dispersion functions of periodic waveguide arrays. According to these properties, the number of sign changes in the recurrence sequence at modal cutoff equals the number of zeros of the dispersion function of the array. A generalized differential form of this equality is developed, which applies along any path in the parameter space. It allows deriving explicit analytical design rules for these arrays. The derived rules reveal the possibility of supporting a specific number of TE or TM modes, which is independent of coupling conditions. Even under strong coupling, it is shown that the zeros of the consecutive recurrence dispersion functions are interlaced. A recurrence zero-search algorithm employs this interlacing in resolving closely-spaced zeros of the dispersion functions of both symmetric and asymmetric arrays. The algorithm is applied with the derived rules in maximizing phase and group birefringence of single-mode silicon-on-insulator (SOI) waveguides. Two strip-loaded SOI waveguides are designed with phase and group birefringence of 1.03 and 1.64 at a free-space wavelength of 1.55 mum.