The field of extreme value statistics is concerned with modeling and predicting rare events. In a Hüsler-Reiss graphical model, a graph represents extremal conditional independence (CI) relations between random variables. These models are exponential families parameterized by a graph Laplacian and are considered the analogue of multivariate Gaussian models in the extremal setting. We study these models from the perspective of algebraic geometry. Translating the CI relations into polynomial constraints in the parameters, we define extremal CI ideals and find a determinantal representation of their generators. In terms of parametric inference, we study the extremal maximum likelihood degree as the number of solutions to a conditionally negative definite matrix completion problem. We also define and analyze the extremal maximum likelihood threshold for Hüsler-Reiss graphical models, which provides a certificate for the existence of a surrogate MLE in terms of the dimensionality of the point configuration that realizes the underlying summary statistic as a Euclidean distance matrix. We highlight throughout many interesting similarities but also differences with respect to Gaussian graphical models.