You could do that, but it doesn't make sense to me, unless you have some kind of weird application.
Doing Signal-Plane-Plane-Signal gives you:
- Excellent high-frequency power delivery thanks to the capacitor formed from the two inner planes (think about it -- two large sheets of copper separated from each other by a thin dielectric)
- Each signal layer has an immediately adjacent reference plane layer, making microstrip transmission line design easy, or more simply, return currents will enjoy a low impedance path barring any weird plane splits
What you've suggested will rob the first item, and create an interesting situation for the second where the stack up would dictate which of the two plane layers your signals will use as a reference plane (based on the distance to each).
You can always do a large flood pour of GND if you need it for some reason (thermal?) on the bottom layer, but unless you have a compelling reason, I'd stick with SPPS as the four layer stack up of choice.
I have read briefly about PSSP stackups, but I was not fully convinced by the few sentences that claimed improved EMI performance by burying all the signals inside; while it's true that solid plane layers would probably help in that regard, I personally don't think it'd be worth the routing hit -- you would have to via everything.