Phillips 66 and activist investor Elliott face off on a classic conglomerate quandary: Are the pieces of a huge company worth more together or broken up?

• The contest pits one of the energy sector’s most storied players, which started 108 years ago as the Phillips Petroleum Co., against arguably the most influential activist fund manager in the world, Elliott Management, led by billionaire Paul Singer.

If you happened upon the website Streamline66.com, you'd find a slickly designed site devoted to a company in crisis, complete with a plea to shareholders to step in and rescue a company in "urgent need of a new direction."...