Excellent, if you can achieve that.
Some people say that the pistonic motion of the voice coil pushes the cone forward (and backward), which in turn pushes the air molecules in front of the cone, causing them to push the next molecule, and so on, allowing our ears to perceive that sound. The question is, when the cone moves back, does it pull the air backward, meaning that no air molecules are pushed outward during that movement? When the cone moves backward, its convex side seems to push air outward into the speaker box. If this is the case, there is a moment when sound is emitted from the cone, followed by a moment when sound moves in the opposite direction not emitting any forward sound, albeit with a very small time difference.
The strange thing is that convex sides can't effectively push air unless there is a tube around them; otherwise, they will simply slide through the air (arrow for example). Alternatively, the convex back side of the cone can push air perpendicular to its surface, meaning that sound (air molecules) won't be directed towards the back plate but rather towards the corners, assuming we consider the idea of air pushing the molecules.
So how does the cardioid work?
This is a very interesting way to formulate a question around this topic. I think it is crucial to first understand why waves in the air can propagate to begin with, and generally how gasses work.
Air is a mix of gasses, so for simplicity we can think of it as a gas, and use the therm "air molecules" or even simpler: "air particles" as a basis for this. The air is not packed full of particles. The density is allways a function of the pressure and temperature of the gas. Both of them can be described as the energy contained within the gas. The energy that comes from pressure can be viewed as a compressed spring. When released, its energy is also released. The heat contained inside the gas is very similar. But when heat determines the speed of air particles bouncing into each other randomly, the pressure is a sort of combination of the heat (speed) and the density (the number of air particles that hits in a given amount of time). So what keeps the gas from collapsing in on itself is simply all the particles moving in all directions and bouncing into each other.
We can imagine holding up a piece of paper in the air. An A4 sheet will have almost 620kg of push from air particles bouncing off on each side all the time. Since the two sides experience the same force and the particles hit the paper in all possible directions, everything will cancel out to a net zero force on the sheet of paper.
It is kind of the same thing with a diapragm. As we move the diaphragm a tiny amount, we make sure we move towards the bouncing particles on one side, making them bounche slightly more quickly back towards other particles, resulting in a wave like you desribed above. But it is important to understand that since these particles move in any direction, they will also bounce of the diaphragm in any direction.
When the diaphragm moves the other way, more space will be created for the moving particles. This is means more space will be created for the particles that were supposed to hit these particles, resulting in more space for the next layer, and so on. So the over-pressure and the under-pressure waves propagate the exact same way.
To further improve the understanding of this, it is important to understand that an object moving at constant speed through air does not create much sound, and hardly any pressure besides a local pressure front. A loudspeaker diaphragm creates pressure by accellerating, not by moving. When we apply current to the voice coil, a force is created. And as we all know, a force makes a mass accellerate. The voltage peak in the signal results in a current peak in the motor, that translates to an accelleration peak in the diaphragm. So while it seems intuitively correct to think of peak displacement of a diaphragm as the signal amplitude, it is actually the peak accelleration of the diaphragm that is the peak amplitude, and also the peak pressure.
This means that viewing a diaphragm as a "snow plow" in air is not the correct way to describe how sound is generated. The shape of the diaphragm does affect how sound spreads, but not because it generates a plow edge, rather because when you measure or hear the sound emitted, it does make a difference where in space different parts of the sound wave emits from. If you measure the sound emitted at an angle from a large cone, you will see that the center and the edge is more in phase if you measure the back than if you measure the front of the cone.
This also means the driver mounted in a cardioid box will in theory load the side ports more effectively when mounted the right way, than it would if you flip it around.
