If you want to get actual propulsion out of the double, and not just go through the motions, extend your leg with each underpush. See how Eddy nearly straightens his leg with the double push, twice per leg before the other skate takes its turn.
Because the underpushing skate travels laterally and forward, it doesn't pop up Eddy's body all that much. A big lateral underpush causes severe instability, which could cause Eddy to topple over if he didn't whip his skate back under him to catch his weight in preparation for a huge old honking regular push.
So why not honk both pushes? Dedicated to Daniel Dugan and Autumn Marsh.