People excel at attributing intentionality to other agents, whether in simple scenarios such as shapes moving in two dimensions or complex scenarios such as people interacting. We note that intentionality attributions seem to fall into two categories: low-level intentionality in which an observer has a theory of mind about an agent, and high-level intentionality in which an observer believes the agent has a theory of mind about something else. We introduce the terms L-intentionality and H-intentionality to refer to these attributions, respectively, and describe this division by using examples from previous research. Social robots provide a particularly good platform for evaluating the presence of different types of intentionality, and we discuss how robots can help distinguish the relationship between H- and L-intentionality, based on a number of possible models that we enumerate. We conclude by highlighting some interesting questions about intentionality in general and the interplay between H- and L-intentionality in particular.