On lossless isotropic planar waveguides the discrete proper modes of propagation form independent transverse electric and transverse magnetic sets such that there is no mode coupling or interaction between modes. In the event of material loss or gain, mode interactions are possible, leading to a complicated spectrum and apparent nonuniqueness of the modes. In this paper we analyze for the first time the cause of these modal interactions by studying the simplest canonical planar waveguide which exhibits these effects, the symmetric-slab waveguide. We show that mode interactions are due to the migration of complex-frequency-plane branch points associated with specific wave phenomena, with varying loss or gain. As these singularities move near the real-frequency axis they influence the modal behavior for time-harmonic (real-valued) frequencies, crossing the real axis at some critical value of loss or gain. It is shown that as time-harmonic frequency varies, passing above, below, or through these branch points results in different modal behavior. Passing above or below, and near to, the branch point yields mode coupling behavior, while passing through the branch point results in modal degeneracy. The result of this branch point migration is that the association of a particular mode with a certain branch of the dispersion function depends not only on the value of material loss or gain, but also on the order in which physical parameters of the problem are varied. Three different branch point types are identified and discussed, which leads to an understanding of the relevant wave phenomena and to a method for organizing the mode spectrum in a consistent and unique manner. While many of the observations described here are based on careful numerical analysis of the transverse magnetic modes existing on a certain symmetric-slab waveguide, the described phenomena are reasonably expected to be generally found in other open dielectric waveguiding structures.