The story on the twisted sticks, known as a culebra (spanish for "snake") is actually not quite accurate.
The torcedors were allowed to smoke and take home a few sticks each day, but they were supposed to smoke/take home only specific designated "lower cost" cigars. Too many workers were ignoring the low cost sticks and taking the pricey stuff to smoke. The powers-that-be decided they had to stop this practice. They developed the culebra for the workers. If they were caught smoking anything but one of these - they were busted.
Many factories had culebras at one time, but they are now made by very few, Drew Estates being one that makes them available for sale.