A B737-800 Captain reported both transponders were not operating after takeoff. He elected to return to departure airport even though ATC seemed amenable to him continuing with no transponder.

Narrative

After departure it became apparent that our transponders were not functioning properly. The bottom line was; both transponders were sending a code of 1200 even though the correct squawk was selected on the unit. The altitude reporting function was working correctly. Many questions came to mind as we climbed out enroute to DCA. Was I legal to enter controlled airspace? Was I legal to enter RVSM airspace? TRACON and Center had no problem with allowing me to continue; but what would Potomac Approach think about it? I contacted Dispatch to ask that our ATC liaison contact FAA ATC with that question. Very uncomfortable with the situation; while climbing out; Center informed me that my transponder info was about 10 miles behind my primary target. The Controller went on to say that upon reaching our cruise altitude they would have me turn off my transponder! With that info; I elected to return for repairs. I did not feel safe to continue the flight into RVSM airspace; with no transponder; no TCAS; as well as operate in the dense traffic environment found in the DC area; as well as the possibility of encountering security issues. I was surprised that ATC would allow me to do so!