Once one head's local operative gets into re-election trouble, then big Geryon's money flows in bountiful measure into the coffers of a seat contender from the other head.
Think this simple rule through and you'll see how well it works.
One result: if a seated head's agent is in a pickle (like us lefties could very easily put dear Jane in) -- once there's blood in the water, it's survival politics at its most naked. The incumbent's party, like a reef shark in a feeding frenzy, will devour its own creature, allowing it to be replaced by a flea from the hair of Orthrus' other head. (I know, mixed metaphor, but screw it, it's a goddam blog.)
Better to let the other team have a turn, than allow some upstarts to disrupt the machinery.
Of course, the weaker head can never be permitted to die out completely, because without a second head, the con is finished.
In this light, somebody ought to trace the tick-tock of the heads in the people's house since, say, the civil war.