That's pretty wacky behavior. I know how the system can do that, but I don't know why it would.
Here is what is happening. The monovalve is opening at the same time as the diverter flap. Does that make it perfectly clear?

Seriously, the monovalve controls the flow of hot water into the heater core. The hot air makes is obvious the monovalve is open. The question is why. Either the Pushbutton Unit (PBU) is telling it to open (which would be a fault in the PBU) or the monovalve itself isn't working correctly.
The diverter valve opens to permit air which has passed through the a/c evaporater to come out the center vents without passing through the heater core. That's why the center vent air is cold, but the side vent air is hot - the side vent air passes through the a/c evap first and then the heater core second.
What's odd is that the diverter valve isn't supposed to open unless it is *really* stinkin hot inside the car - like when it's been sitting in the sun and the inside temperature is 130. I would never expect the diverter to open on a mild day while driving down the highway.
This is a difficult problem to diagnose based on the symptoms. I'm leaning toward convicting the PBU. However, what I think you really need is someone who knows these systems to go through it and test all the sensors and components individually. I have a suspicion there is more than one thing going on here. I wouldn't run out and spend the $$ on a PBU as trial and error on this one.
My $.02,
-JimY