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Two main scenarios have been proposed for origin of massive runaway stars -- dynamical ejection or release from a binary at the first core collapse -- but their relative contribution remains debated. Using two large spectroscopic campaigns towards massive stars in 30 Doradus, we aim to provide observational constraints on the properties of the O-type runaway population in the most massive active star-forming region in the Local group. We use RV measurements of the O-type star populations in 30 Doradus obtained by the VLT-FLAMES Tarantula Survey and the Tarantula Massive Binary Monitoring to identify single and binary O-type runaways. We discuss their rotational properties and qualitatively compare observations with expectations of ejection scenarios. We identify 23 single and one binary O-type runaway objects, most of them outside the main star-forming regions in 30 Doradus. We find an overabundance of rapid rotators (vsini > 200km/s) among the runaway population, providing an explanation of the overabundance of rapidly rotating stars in the 30 Doradus field. Considerations of the projected rotation rates and runaway line-of-sight (los) velocities reveal a conspicuous absence of rapidly rotating (vsini > 210k/ms), fast moving (v_{los} > 60km/s) runaways, and suggest the presence of two different populations of runaway stars: a population of rapidly-spinning but slowly moving runaways and a population of fast moving but slowly rotating ones. These are detected with a ratio close to 2:1 in our sample. We argue that slowly moving but rapidly spinning runaways result from binary ejections, while rapidly moving but slowly spinning runaways could result from dynamical ejections. Given that detection biases will more strongly impact the slow-moving population, our results suggest that the binary evolution scenario dominates the current massive runaway population in 30 Doradus.