Human interactions are organized in sequence, which is a key component of Levinson's "interaction engine." Referring back to the field where it originated, Conversation Analysis, we discuss its relevance within the interaction engine, before moving on to show how sequence organization is actually oriented to not only humans in social interaction, but also to non-human animals. On the basis of video-recorded encounters between baboons (Papio Anubis), we study canonical sequences constituting openings and, within them, greetings. Openings are the locus where future interactants adjust to each other to coordinately enter in interaction, thus achieving a common definition of their context, activity, and relations. The analysis shows that the ways individuals spatially approach each other provide systematic interactional affordances for how the first sequences of actions in the opening are formatted, initiated, and responded to. Adopting sequential multimodal analysis, we demonstrate how participants orient to central features of sequence organization-its sequential implicativeness and the expectations it produces-building on them their interpretations of others' actions, their responsivity, and their mutual understanding of the ongoing course of action as it unfolds. This paves the way for further reflections on the pervasiveness of the interactional engine in human and non-human primate communication.