Empirically derived kappa distributions are becoming increasingly widespread in space physics as the power law nature of various suprathermal tails is melded with more classical quasi-Maxwellian cores. Two different mathematical definitions of kappa distributions are commonly used and various authors characterize the power law nature of suprathermal tails in different ways. In this study we examine how kappa distributions arise naturally from Tsallis statistical mechanics, which provides a solid theoretical basis for describing and analyzing complex systems out of equilibrium. This analysis exposes the possible values of kappa, which are strictly limited to certain ranges. We also develop the concept of temperature out of equilibrium, which differs significantly from the classical equilibrium temperature. This analysis clarifies which of the kappa distributions has primacy and, using this distribution, the kinetic and physical temperatures become one, both in and out of equilibrium. Finally, we extract the general relation between both types of kappa distributions and the spectral indices commonly used to parameterize space plasmas. With this relation, it is straightforward to compare both spectral indices from various space physics observations, models, and theoretical studies that use kappa distributions on a consistent footing that minimizes the chances for misinterpretation and error. Now that the connection is complete between empirically derived kappa distributions and Tsallis statistical mechanics, the full strength and capability of Tsallis statistical tools are available to the space physics community for analyzing and understanding the kappa-like properties of the various particle and energy distributions observed in space.