Cool reds will mix with blues to create clean purples, with yellows to create mid to neutral oranges and with green to create greys. Cool reds include Alizarin Crimson*, Permanent Alizarin Crimson Hue, Spectrum Crimson, Anthraquinoid Red, Permanent Carmine, Permanent Rose, Quinacridone Rose and many others. Mixed together, the warm red and cool red will create a lovely pure red that is neither orange nor purple biased.
In a split primary palette is it useful to have one of each type of red to create clean colours. The cool reds are those that lean to the purple side, rather than the yellow side.