Easiest way to understand it is to imagine the polar pattern of a figure-8 microphone.
A figure-8 microphone has two circular lobes positioned fore and aft of the capsule. Fore of the capsule is "in phase" and the aft lobe is "out of phase".
Figure-8 microphones sense pressure gradient information, which is the rate of change of acoustic ...
The classic way of creating cardioid pickup pattern was to couple an omni with a figure 8 capsule, where the + side of the figure 8 is pointing forward.
So I guess in theory you could add 4 figure 8 mics to the setup, and end up with cardioid pickup patterns, but I'm not sure if this would be the most cost effective solution.