CAPT OF A B737 TURNED AND CLBED IN RESPONSE TO NEW HEADING GIVEN BY APCH CTL AND SUBSEQUENT TCASII RA DURING VECTORS TO SFO ILS. BOTH ACFT WERE NOSE TO NOSE TURNING FINAL FROM OPPOSITE BASED LEGS.

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WE WERE BEING VECTORED ON A 120 DEG HDG AT 6000 FT (L DOWNWIND ILS RWY 28 SFO). OUR NEXT TURN WAS TO A 030 DEG HDG (AN APPARENT L BASE). ALMOST IMMEDIATELY AN ACFT APPEARED ON THE TCASII ON AN APPARENT R BASE FOR RWY 28R AT 6000 FT. OUR PATHS WOULD CONVERSE AT ABOUT THE SAME SPOT ON THE LOC IF NEITHER WAS GIVEN A TURN. ABOUT THE TIME I WAS GOING TO QUERY ATC ON OUR SIT 2 THINGS HAPPENED ALMOST SIMULTANEOUSLY. ATC ASKED US OUR HEADING STATING THAT WE HAD AN ACFT AT OUR ALT AND WE RECEIVED A TCASII TA. THE FO STATED OUR HEADING WAS 030 DEGS. ATC GAVE US AN IMMEDIATE R TURN TO 100 DEGS (APPROX HEADING) AND ALSO GAVE THE OTHER ACFT A R TURN AWAY FROM US. AS WE WERE IN THE TURN WE RECEIVED A TCASII RA TO 'CLB' WHICH I FOLLOWED. AT ABOUT 6500 FT WE RECEIVED A CLR OF CONFLICT RPT FROM TCASII. AS WE BEGAN THE TCASII CLB THE FO RPTED TO ATC THAT WE WERE IN A TCASII CLB. ATC RESPONDED ASKING US WHAT OUR HEADING HAD BEEN. THE FO STATED 030 DEGS. THEN ATC ASKED FOR OUR HEADING BEFORE THAT TURN. THE FO STATED 120 DEGS. ATC THEN GAVE US A L TURN TO A 280 DEG HDG. I ASKED THE FO TO VERIFY THAT THAT WAS A L TURN TO 280 DEGS. ATC RESPONDED TURN L TO 280 DEGS; INTERCEPT THE RWY 28R LOC; MAINTAIN 4000 FT TO DUMRA; CLRED FOR THE ILS TO RWY 28R. NO FURTHER MENTION WAS MADE BY ATC REGARDING THE MIX-UP. AFTER LNDG I CALLED BAY APCH ON THE TELEPHONE TO SEE IF THEY NEEDED TO TALK TO US. THE GENTLEMAN THAT ANSWERED SAID HE'D CHK. WHEN HE CAME BACK HE INDICATED THAT THEY WOULD HAVE TO INVESTIGATE TO SEE IF REQUIRED SEPARATION HAD BEEN MAINTAINED. HE ASKED IF I THOUGHT WE'D GOTTEN CLOSER THAN 3 MI. I SAID I DIDN'T KNOW; BUT THAT WE WERE IN THE TURN BEFORE RECEIVING THE TCASII RA. HE SAID 'THAT'S GOOD.' HE SAID THAT THEY DIDN'T HAVE A PROB WITH US; BUT THAT THE CTLR THOUGHT HE'D GIVEN US A 330 DEG HDG NOT 030 DEGS. HE WENT ON TO SAY THAT EVEN IF WE TURNED TO AND READ BACK THE WRONG HEADING; THE PROB THEY HAD WAS WHETHER THE CTLR HAD MAINTAINED ADEQUATE SEPARATION. I ASKED IF THAT MEANT WE COULD EXPECT ANY ACTION FROM THEM CONCERNING OUR PART AND HE SAID 'NO' THEY WOULD BE INVESTIGATING THE CTLR'S ACTIONS NOT OURS. THAT ENDED THE PHONE CALL.