B737-700 FLT CREW IS GIVEN CONFUSING ATC INSTRUCTIONS WHILE ESTABLISHED ON FINAL APCH. WHEN CLRNC IS FOLLOWED; A TCAS RA IS ENCOUNTERED.

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WHILE ON FINAL TO RWY 24R AT LAX; JUST INSIDE OF PALAC AND AFTER BEING CLRED FOR THE ILS RWY 24R BY THE PREVIOUS CTLR THE (NEW) FINAL CTLR SUBSEQUENTLY ISSUED US; 'COMPANY NUMBER TURN R; FLY HDG 160.' TURNING R FROM THE 249 DEG FINAL APCH COURSE FOR RWY 24R TO A HDG OF 160 DEGS IS A R 270 DEG TURN. THIS CLRNC DID NOT MAKE SENSE TO US AS IT WOULD PUT US IN CONFLICT WITH OTHER INBOUND TFC TO THE 24 COMPLEX AND BECAUSE WE HAD ALREADY BEEN CLRED FOR THE ILS. WE ATTEMPTED TO VERIFY THE CLRNC WITH THE CTLR. WE WERE UNABLE TO VERIFY THE CLRNC DUE TO HVY CONGESTION ON THE FREQ SO I CONTINUED TRACKING THE LOC AND GS. THE FO TRIED A SECOND TIME TO CONFIRM THE CLRNC AND WAS AGAIN BLOCKED BY ANOTHER ACFT ON FREQ. AT THIS POINT I BEGAN A R TURN TO COMPLY WITH THE CLRNC. ON THE FO'S THIRD ATTEMPT HE WAS ABLE TO QUERY THE CTLR. THE CTLR SAID; 'NEGATIVE COMPANY NUMBER TURN L HDG 210; MAINTAIN 5000 EXPECT 24L.' BY THIS TIME; WE WERE THOROUGHLY CONFUSED AND FRUSTRATED. WE WERE ORIGINALLY CLRED FOR THE ILS RWY 24R; WE WERE THEN GIVEN A CLRNC OF R TURN TO HDG 160 DEGS (WITH NO REASON WHY) AND NOW WE WERE GIVEN A RWY CHANGE FROM RWY 24R TO RWY 24L. WE TRIED TO CONFIRM THIS CLRNC TO RWY 24L BUT WE RECEIVED A CLB RA AS THE FO WAS KEYING THE MIKE. I DISCONNECTED THE AUTOPLT AND COMPLIED WITH THE RA. NOW THE CTLR WAS YELLING AT US AND ASKING WHAT WE WERE DOING. THE FO TOLD HIM WE WERE RESPONDING TO AN RA. THE CTLR THEN SAID WE WERE TAKING CLRNCS FOR ANOTHER CARRIER WITH THE SAME FLT NUMBER AS OURS. I KEYED THE MIKE AND SAID HE HAD GIVEN US 3 CLRNCS FOR COMPANY NUMBER. ANYTHING IS POSSIBLE OF COURSE AND OBVIOUSLY I HAVE NOT HEARD THE ATC TAPES; BUT I FIND IT HIGHLY UNLIKELY THAT WE WERE TAKING CLRNCS FOR ANOTHER ACFT. THE CTLR ISSUED INSTRUCTIONS FOR COMPANY NUMBER 3 SEPARATE TIMES AND BOTH MY FO AND I BOTH HEARD 'COMPANY NUMBER' ALL 3 TIMES. FURTHER; FO READ BACK 'COMPANY NUMBER' 3 TIMES IN RESPONSE TO THE CLRNCS AND THE CTLR DID NOT CORRECT IT. ADDITIONALLY; NEITHER I NOR FO EVEN HEARD ANOTHER CARRIER FLT NUMBER SAME AS OURS ON FREQ. I ALSO QUESTION WHY WE RECEIVED AN RA. HAD WE BEEN ABLE TO QUICKLY VERIFY THE CLRNC WITH ATC; THERE NEVER WOULD HAVE BEEN AN ISSUE. IF IN FACT THERE WAS A SIMILAR CALL SIGN ON FREQ; A HEADS UP WOULD HAVE HELPED.