The core belief of this religion is that the twine ball will somehow defeat Godzilla and allow the aliens from Planet X to come to Earth safely. Various interpretations of this core belief form the three separate Twiner denominations.
PHILOSOPHICAL TWINERS believe the twine ball is a metaphorical representation of peace, much as Godzilla was originally a metaphor for the devastation of war, mankind's aggressive nature, and nuclear fallout. These people believe the aliens are waiting until mankind has evolved beyond it's warlike nature at which time they will safely make contact with us.
FUNDAMENTALIST TWINERS believe the twine ball will literally hatch a giant monster which will defeat Godzilla, thus allowing the aliens to safely land on Earth.
PESSIMISTIC TWINERS believe the twine ball will hatch a giant monster which will promptly be eaten by Godzilla like every other monster before it making the entire religion an exercise in futility.