Blue-based reds fall in the alizarin family. And orange/yellow based reds fall in the vermillion family.

Alizarin is here: http://en.wikipedia.org/wiki/Alizarin_crimson_%28color%29

But if you scroll down, you can see some of the other reds as well.

It's also true of blues (cool and warm blue), yellows, and blacks (red-based black, green or blue and so on). What can make disparate colours compatible is sometimes the root colours behind a specific shade.