This image shows a quad-breaker with handle ties.
Image borrowed from this DIY question
One of the comments noted that despite the "handle ties", this is not a "common trip" breaker. I, in my ignorance, previously assumed that a trip on one of the tied breakers would have forced the tied mate to trip as well and that this would, therefore, also be a "common trip". It seems I was wrong.
What is the difference between a "handle tie" and a "common trip"? For bonus points, when should each be used?
As was pointed out in an answer on that original question, this quad-breaker is labeled
ALL POLES HAVE
NO COMMON TRIP
Does this mean that "common trip" is internal to the breaker and can only be identified by reading the labeling, while "handle tie" is the external plastic/metal piece that physically ties the handles together? If so, does that mean (theoretically) that one could have a common trip breaker without a handle tie?