Heatwaves are divided between moderate, more common heatwaves and rare "high-mortality" heatwaves that have extremely large health effects per day, which we define as heatwaves with a 20% or higher increase in mortality risk. Better projections of the expected frequency of and exposure to these separate types of heatwaves could help communities optimize heat mitigation and response plans and gauge the potential benefits of limiting climate change. Whether a heatwave is high-mortality or moderate could depend on multiple heatwave characteristics, including intensity, length, and timing. We created heatwave classification models using a heatwave training dataset created using recent (1987-2005) health and weather data from 82 large US urban communities. We built twenty potential classification models and used Monte Carlo cross-validations to evaluate these models. We ultimately identified several models that can adequately classify high-mortality heatwaves. These models can be used to project future trends in high-mortality heatwaves under different scenarios of a changing future (e.g., climate change, population change). Further, these models are novel in the way they allow exploration of different scenarios of adaptation to heat, as they include, as predictive variables, heatwave characteristics that are measured relative to a community's temperature distribution, allowing different adaptation scenarios to be explored by selecting alternative community temperature distributions. The three selected models have been placed on GitHub for use by other researchers, and we use them in a companion paper to project trends in high-mortality heatwaves under different climate, population, and adaptation scenarios.