DC9 CREW AT FL330 BELIEVED THEY WERE GIVEN A R TURN FOR TFC; THEN CORRECTED BY THE ZID CTLR WITH SEVERAL MORE TURNS L. CTLR CLAIMS THE PLT LOST SEPARATION WITH ANOTHER FLT AT FL330.

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AT APPROX XA00Z WE WERE GIVEN 10 DEG COURSE CORRECTION BY ZID ON FREQ 126.92. I THOUGHT I HEARD 10 DEGS R. WE WERE HDG 090 DEGS DIRECT TO BECKLEY VOR (BKW) AT FL330; SO I TURNED TO A HDG OF 100 DEGS. APPROX 2 MINS LATER THE CTLR ASKED OUR HDG. WE RESPONDED 100 DEGS. SHE THEN TOLD US TO TURN 20 DEGS L TO A HDG OF 080 DEGS FOLLOWED QUICKLY BY A FURTHER L TURN TO 180 DEGS. PASSING ABOUT 250 DEGS SHE CLRED US BACK TO THE R DIRECT TO BKW AND GAVE US A PHONE NUMBER TO ZID QUALITY ASSURANCE; ASKING US TO CALL WHEN WE GOT ON THE GND. AFTER I LANDED AT DCA; I CALLED AND SPOKE WITH MR X AT ABOUT XB15Z AND HE TOLD ME HE WAS JUST NOTIFIED OF THE SIT AND TO CALL BACK IN 1 HR. I CALLED BACK AND HE SAID THEY HAD PULLED THE TAPES AND AS A RESULT OF OUR INITIAL TURN THE CTLR HAD LOST MINIMUM SEPARATION WITH A SBOUND ACR Y ACFT AT FL330. HE SAID WE WERE 4.8 MI FROM ACR Y. HE SAID THE LOSS OF SEPARATION ALARM WENT OFF AND HE WOULD HAVE TO GENERATE PAPERWORK. HE SAID THAT ON THE TAPE THE CTLR TOLD US TO TURN L 10 DEGS; WE READ BACK L 10 DEGS; BUT IT APPEARED TO HER WE TURNED R SINCE SHE HAD US HDG 090 DEGS; AND WHEN SHE ASKED OUR HDG BEFORE THE SECOND TURN WE RESPONDED 100 DEGS. HE RESTATED WE GOT WITHIN 4.8 MI OF ACR Y. THE ACR Y ACFT NEVER SHOWED UP ON OUR TCAS; WE NEVER HAD THE ACFT VISUALLY AND NO EVASIVE MANEUVERS WERE PERFORMED. ULTIMATELY; IT APPEARS I MAY HAVE TURNED R INSTEAD OF L; SO IT IS IMPERATIVE I CONFIRM THE DIRECTION OF TURN WITH THE NEW HDG AS THIS MIGHT HAVE INDICATED TO THE CTLR QUICKER THAT I TURNED THE WRONG DIRECTION. I ALSO THINK IF SEPARATION IS BECOMING A FACTOR; THE CTLR SHOULD ASK OUR HDG; THEN GIVE A DIRECTION OF TURN AND STATE THE NEW HDG THEY WANT US ON. THIS WILL HELP ALSO TO CLR UP CONFUSION. WE WERE AT CRUISE ALT AND WORKLOAD WAS LIGHT; I JUST MISUNDERSTOOD AND TURNED R INSTEAD OF L. IF WE WERE CO-ALT AT 4.8 MI AWAY; I AM AT A LOSS AS TO WHY THE TCAS DID NOT ALERT ME OR EVEN SHOW UP AS A TARGET ON THE OTHER ACFT.