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We study the spatiotemporal correlation of terrorist attacks by al-Qaeda, the Islamic State of Iraq and Syria (ISIS), and local insurgents, in six geographical areas identified via k-means clustering applied to the Global Terrorism Database. All surveyed organizations exhibit near-repeat activity whereby a prior attack increases the likelihood of a subsequent one by the same group within 20 km and on average 4 (al-Qaeda) to 10 (ISIS) weeks. Near-response activity, whereby an attack by a given organization elicits further attacks from a different one, is found to depend on the adversarial, neutral, or collaborative relationship between the two. When in conflict, local insurgents respond quickly to attacks by global terror groups while global terror groups delay their responses to local insurgents, leading to an asymmetric dynamic. When neutral or allied, attacks by one group enhance the response likelihood of the other, regardless of hierarchy. These trends arise consistently in all clusters for which data are available. Government intervention and spillover effects are also discussed; we find no evidence of outbidding. Understanding the regional dynamics of terrorism may be greatly beneficial in policy making and intervention design.