Context. The origin of massive runaway stars is an important unsolved problem in astrophysics. Two main scenarios have been proposed, namely: dynamical ejection or release from a binary at the first core collapse. However, their relative contribution remains heavily debated. Aims. Taking advantage of 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. Methods. We used radial velocity 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. Here, we discuss the rotational properties of the detected runaways and qualitatively compare the observations with expectations of ejection scenarios. Results. We identified 23 single and one binary O-type runaway objects, most of them located outside the main star-forming regions in 30 Doradus. We find an overabundance of rapid rotators (ve sin i > 200 km s−1) among the runaway population, thus providing an explanation for the observed overabundance of rapidly rotating stars in the 30 Doradus field. Considerations of the projected rotation rates and runaway line-of-sight velocities reveal a conspicuous absence of rapidly rotating (ve sin i > 210 km s−1), fast-moving (vlos > 60 km s−1) runaway stars in our sample, strongly suggesting the presence of two different populations of runaway stars: a population of rapidly spinning but slowly moving runaway stars and a population of fast-moving but slowly rotating ones. These are detected with a ratio close to 2:1 in our sample. Conclusions. We argue that slowly moving but rapidly spinning runaway stars 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 runaway population, our results suggest that the binary evolution scenario dominates the current massive runaway star population in 30 Doradus.