Internet memes are a popular and long-standing genre of discourse on social media platforms, used to express everything from emotional states to political opinions. Dancygier and Vandelanotte (2017) define internet memes as intertextual, multimodal discourses that combine text with images. In order to capture and compare these rapidly-changing discourses, we propose a descriptive dual classification system for memes with two components: meme composition and multimodal quality. Meme composition categorizes memes by their structure—beyond the individual images they employ—and thus explains how memes recontextualize images and text to create new meanings. Multimodal quality serves to describe the way(s) that the text interacts with the image in the meme: as a caption, label, and/or utterance. Combining one meme composition with one or more multimodal qualities classifies an individual meme structurally and provides a basis for explaining its intertextuality, modality, and meaning-making. We apply the dual classification system to English language data collected in its naturally-occurring context on the social media platform Instagram from 2019 to 2021. Analysis of these data shows that the dual classification system is a flexible and robust approach which provides a vocabulary for discussing the creative agency exerted by meme creators in a wide range of communities. We argue that the dual classification system affords researchers the ability to study memes linguistically across a variety of platforms and over time.