IFR B190 INBOUND TO IAD IS IN DISAGREEMENT WITH ATC REGARDING ASSIGNED HEADING.

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WE WERE ON AN ASSIGNED RTE OF CARAS 4000 FT DEPART CARAS 190 EXPECT RWY 1R. THE CTLR HANDED US OFF TO APCH. WE CHKED IN WITH ASSIGNED ALT. THE CTLR GAVE US A HEADING OF 340 DEGS EXPECT RWY 1L. I READ BACK THE CLRNC 340 DEGS; 4000 FT AND SAID YOUR CHANGING MY RWY? THERE WAS NO REPLY. ASKED FO IF HE HEARD THE SAME THING AND HE REPEATED THE SAME. WE PROCEEDED TO THE ASSIGNED HEADING. WE KNEW THIS WOULD TAKE US OVER THE TOP OF THE ARPT. 1 MIN LATER THE CTLR ASKED 'WHERE ARE YOU GOING?' I REPLIED '340 DEGS ASSIGNED.' HE SAID 'YOU ARE GOING THE WRONG WAY; YOU WERE LISTENING TO ANOTHER CALL SIGN -- LISTEN UP!' I REPLIED 'NO I DID NOT; THIS IS WHAT WE WERE ASSIGNED.' THERE WAS A 30 SECOND DELAY AND WE HAD A NEW CTLR WHO VECTORED US AROUND FOR RWY 1R DULLES WITH NO PROBS. SUPPLEMENTAL INFO FROM ACN 582935: (NEW VOICE) WE KNOW IT IS NEW BECAUSE IT IS NOT MALE BUT A FEMALE CTLR -- WHO CRISPLY AND PROFESSIONALLY VECTORS US AROUND TO FINALLY INTERCEPT THE RWY 1R ILS AT DULLES. LAND UNEVENTFULLY. MY POINT ON THE SAFETY OF FLT ISSUE IS; AS A FORMER MIL FLYER WE WERE TAUGHT; AND I FIRMLY BELIEVE IT IS A GOOD AXIOM AND APPLIES IN THIS INSTANCE: 'WHILE IN THE AIR; WORK AS A TEAM; SETTLE ANY DIFFERENCES ON THE GND.' WITHOUT HEARING THE TAPES OF WHAT TRANSPIRED IN THE COM; THE PIC AND I COULD HAVE HEARD A WRONG CALL SIGN. THE PERCENTAGES OF THIS ARE LOW; BUT POSSIBLE. THE CTLR COULD HAVE MEANT ANOTHER ACFT BUT USED OUR CALL SIGN -- HE'S HUMAN ALSO. HOWEVER; AN ACFT IN IMC CONDITIONS; OVER A BUSY METROPOLITAN ARPT IS NOT A PLACE TO BE ASSIGNING INSTANT BLAME. CALMLY GET US ON THE GND AND CALL ON THE TELEPHONE IF THERE IS A PROB TO SOLVE. I DID NOT FEEL THIS WAS A TOTALLY UNSAFE INCIDENT; THE TCASII SHOWED NO CONFLICTING TFC IN THIS PARTICULAR CASE; BUT THERE COULD HAVE BEEN A PROB IF CONDITIONS WERE DIFFERENT.