Why don't they build..........
 

a rotary engine/electric hybrid? It would seem that this combo would give the best combination of features. The electric motor would provide the low end torque (which the rotary doesn't have), both being "rotary" designs would make the combo very smooth and having the electrics on board would improve the rotary engine's gas mileage. Also, the combined electric motor and the rotary engine would be a very compact assembly and lighter than a similar displacement piston engine hybrid. Am I missing something here or does this make sense?

Yeah, you are missing the fact that rotary engines are not very fuel efficient, and that only Mazda really knows how to build one.

The total rotary action is tempting, few moving parts, etc. Also compact size. Might be good as a on board motor-generator like the Fisker, or others that are always electric driven, hard to beat a modern 3-4 cyl DFI turbo multi valve powerplant though.

this!

I remember when the rotary engine came out (in the '70's?) and thinking that the "normal engine" would be phased out because of too many moving parts. Hearing about fuel consumption explained the lack of popularity of the rotary engine.
I would think with this hybrid combo the fuel consumption hinges on the basis of how much time the rotary engine is running. If if I was driving it the rotary side would be full on speed limits permitting! Nice to dream though.

I think there is a company working on this in Australia. I believe using a hydrogen-powered rotary.

Rotary engines are also stinky. They had to pile a series of inline cats to clean them up.
Just an ICE version of a turbine.

Change the design, run it on CNG or hydrogen, it would have potential.

Poor volumetric efficiency. A banana-shaped combustion chamber will never be as efficient as a round cylinder.

How about a small turbine then? Too much start up and shut down drama?

Turbines are also pollution machines...and got lousy mileage.
Chrysler Turbine Car - Wikipedia, the free encyclopedia

Rotary problems......
 

The reason that rotary engines are not fuel efficient is because they make little torque at low rpms thus requiring a low axle ratio and excessiver engine speed to start moving the vehicle mass. If the electric motor which develops max torque at stall (0 rpms) is used to start the vehilcle moving perhaps up to 20 mph or so, then the rotary can step in and take over from there. By keeping the engine module lightweight, there will be less mass to accelerate or, the battery pack could be made larger to allow the electrics to work more. With some of the load off the rotary, it could very likely be made to run leaner and thus more efficiently. This would also allow for a more simple catalytic converter to be used. I believe that all of the "issues" could be resolved with some careful engineering.

Atkinson cycle piston engine for the suppl. power

One of many companies developing rotary engines for UAVs and they have an automotive range extender version as well.

Cubewano