ACR X NON ADHERENCE TO ATC CLRNC TURN TO WRONG HDG RECEIVED TCASII TA HAD LTSS FROM ACR Y. EVASIVE ACTION TAKEN.

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ACR X DEPARTED RWY 33R; GIVEN 310 DEGS BY TWR. SHORTLY AFTER HAVING CHKED IN WITH DEP AND GETTING 12000 FT; WE WERE THEN GIVEN A TURN TO 210 DEGS AT ABOUT 2000 FT MSL. AFTER ROLLING OUT ON 210 DEGS; NOTICED SEVERAL POTENTIAL UPCOMING CONFLICTS ON TCASII. ALERTED FO. JUST THEN; DEP; IN A CONCERNED VOICE; ASKED OUR HDG. FO SAID 210 DEGS; AS I DID A MAX PERFORMANCE (60 DEG BANK) CLBING R TURN TO AVOID ACR Y; DEPARTING RWY 28. CTLR THEN ASSIGNED 330 DEGS AND ASKED WHY WE WERE NOT ON A 310 DEG HDG. WE SAID BECAUSE SHE HAD ASSIGNED 210 DEGS! SHE SAID SHE DID NOT! GOING TO PULL TAPES. WILL KNOW MORE LATER. FO AND MYSELF BOTH HEARD ACR X AND HDG. BOTH INDEPENDENTLY SET HDG BUGS. BOTH THOUGHT HDG UNUSUAL. BOTH HEARD HIM READ BACK ACR X AND HDG. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING: RPTR STATED HE CONTACTED THE FAA FACILITY TWICE AND WAS TOLD THE TAPES HAD NOT BEEN PLAYED BACK. CONFLICT WAS WITH AN MLG. RPTR STATED HE AND THE FO DISCUSSED THE UNUSUAL HDG BUT DID NOT QUESTION THE CTLR. RPTR SAID THEY MIGHT HAVE TAKEN ANOTHER ACFT CLRNC. RPTR TOOK EVASIVE ACTION WITHOUT RECEIVING A TCASII RA.