© 2019. The American Astronomical Society. All rights reserved. Golovich et al. present an optical imaging and spectroscopic survey of 29 radio relic merging galaxy clusters. In this paper, we study this survey to identify substructure and quantify the dynamics of the mergers. Using a combined photometric and spectroscopic approach, we identify the minimum number of substructures in each system to describe the galaxy populations and estimate the line-of-sight velocity difference between likely merging subclusters. We find that the line-of-sight velocity components of the mergers are typically small compared with the maximum 3D relative velocity (usually <1000 km s-1 and often consistent with zero). We also compare our systems to n-body simulation analogs and estimate the viewing angle of the clean mergers in our ensemble. We find that the median system's separation vector lies within 40° (17°) at a 90% (50%) confidence level. This suggests that the merger axes of these systems are generally in or near the plane of the sky, matching findings in magnetohydrodynamical simulations. In 28 of the 29 systems we identify substructures in the galaxy population aligned with the radio relic(s) and presumed associated merger-induced shock. From this ensemble, we identify eight systems to include in a "gold" sample that is prime for further observation, modeling, and simulation study. Additional papers will present weak-lensing mass maps and dynamical modeling for each merging system, ultimately leading to new insight into a wide range of astrophysical phenomena at some of the largest scales in the universe.