What's the purpose of a dual two-pole breaker?
It is a quad breaker. The outer two are linked and the inner two are linked so that both pairs have 240 V across the connections. They are used to provide two 2-pole breakers, where a pair of ordinary 2-pole breakers would not work because the adjacent half slots are on the same leg.

If the available slots are such that the first pair of half slots are on different legs, then this quad breaker would not be used and would not fit. In that situation one would use a pair of ordinary compact 2-pole breakers.