Thermodynamics don't know where they are being applied - car, house, spacecraft, etc.
The foil goes to the air space side. If there is no air it does not prevent radiant heat transfer. In your case, there is not really an air space but I'd go ahead and put the foil toward the cab. Against the firewall it has zero chance of blocking heat, against the carpet pad or carpet back, at least there is some limited air space for it to function as designed.
Tclederman, in a house the foil side of radiant barrier plywood goes to the attic when used on a roof deck, and towards the outside when used on walls. Both cases provide an air gap at the foil to allow it to work. I've never seen foil facing on insulation unless you're referring to XPS rigid foam sheathing in which case again the foil faces out. But you're in NY and I'm in TX, and as you said, different climates use different methods, either way, the physics are the same.
