Among the many lines of research that have been exploring how embodiment contributes to cognition, one focuses on how the neural substrates of language may be shared, or at least closely coupled, with those of action. This paper revisits a particular proposal that has received considerable attention-namely, that the forms of hierarchical sequencing that characterize both linguistic syntax and goal-directed action are underpinned partly by common mechanisms in left Brodmann area (BA) 44, a cortical region that is not only classically regarded as part of Broca's area, but is also a core component of the human Mirror Neuron System. First, a recent multi-participant, multi-round debate about this proposal is summarized together with some other relevant findings. This review reveals that while the proposal is supported by a variety of theoretical arguments and empirical results, it still faces several challenges. Next, a narrower application of the proposal is discussed, specifically involving the basic word order of subject (S), object (O), and verb (V) in simple transitive clauses. Most languages are either SOV or SVO, and, building on prior work, it is argued that these strong syntactic tendencies derive from how left BA44 represents the sequential-hierarchical structure of goal-directed actions. Finally, with the aim of clarifying what it might mean for syntax and action to have "common" neural mechanisms in left BA44, two different versions of the main proposal are distinguished. Hypothesis 1 states that the very same neural mechanisms in left BA44 subserve some aspects of hierarchical sequencing for syntax and action, whereas Hypothesis 2 states that anatomically distinct but functionally parallel neural mechanisms in left BA44 subserve some aspects of hierarchical sequencing for syntax and action. Although these two hypotheses make different predictions, at this point neither one has significantly more explanatory power than the other, and further research is needed to elaborate and test them.