Fill a mixing glass with ice and add bourbon, vermouth, Campari and blood orange juice. Stir until well combined and very cold.
Strain into a chilled tumbler or rocks glass with fresh ice or serve "up" (without ice). Garnish with blood orange slices or twists of orange peel.
Alternately, you can build the cocktail directly in the serving glass by pouring each ingredient over ice and stirring to mix. If using this method, strain the pulp from the blood orange juice before adding it to the cocktail.