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Very recently, the classification of Moufang polygons has been completed by Tits and Weiss. Moufang $n$-gons exist for $n \in \{3, 4, 6, 8 \}$ only. For $n \in \{3, 6, 8 \}$, the proof is nicely divided into two parts: first, it is shown that a Moufang $n$-gon can be parametrized by a certain interesting algebraic structure, and secondly, these algebraic structures are classified. The classification of Moufang quadrangles $(n=4)$ is not organized in this way due to the absence of a suitable algebraic structure. The goal of this article is to present such a uniform algebraic structure for Moufang quadrangles, and to classify these structures without referring back to the original Moufang quadrangles from which they arise, thereby also providing a new proof for the classification of Moufang quadrangles, which does consist of the division into these two parts. We hope that these algebraic structures will prove to be interesting in their own right.