CONFLICTING ATC INSTRUCTIONS WITH REF TO TCASII RA ON OPPOSITE DIRECTION DSNDING TFC.

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WE WERE NAVIGATING ON J146 (54 DEGS) AT FL290. WE REQUESTED FL330. ATC (ZLA) GAVE US A 010 DEG HDG AND CLB TO FL330. THE FO ACKNOWLEDGED. THEN ATC SAID TO MAINTAIN 1500 FPM OR GREATER. I SET THE CLB EPR AND THE CAPT ESTABLISHED A CLB. SHORTLY AFTER THE CLB WAS ESTABLISHED THE TCASII ISSUED A TA 'TFC;' FOLLOWED BY A RA 'DSND; DSND.' THE CAPT SLOWLY AND SMOOTHLY STOPPED THE CLB; AND BEGAN A DSCNT. I SUGGESTED TO THE FO TO TELL ATC THAT WE WERE FOLLOWING AN 'RA;' WHILE AT THE SAME TIME; ATC INSTRUCTED US TO TURN TO A 340 DEG HDG (IMMEDIATELY). WHILE IN THE TURN TO 340 DEGS; THE FO PICKED UP VISUAL CONTACT WITH THE OTHER ACFT; A 737; WHEN ON THE 340 DEG HDG THE TCASII ISSUED 'CLR OF CONFLICT' AND WE RESUMED CLB TO FL330. I FEEL THAT THE PRIMARY CAUSE OF THIS INCIDENT WAS THE CTLR'S GIVING US A CLB CLRNC WITH INADEQUATE SPACE WITH A DSNDING ACFT. THIS MAY HAVE BEEN A 'REAL' SPACE PROB OR ONE PERCEIVED BY THE TCASII UNIT. COMPOUNDING THE PROB WAS CONFLICTING; CONFUSING INFO. ATC WAS INSTRUCTING US TO CLB; WHILE THE TCASII WAS INSTRUCTING US TO DSND. THERE WAS ALSO THE UNCERTAINTY OF WHAT ACTION (IF ANY) THE OTHER ACFT MIGHT TAKE. OUR REACTION TO THIS (TCASII RA) WAS PROBABLY SLOWED A BIT DUE TO THIS CONFLICTING INFO AND OUR ATTEMPTS TO EVALUATE IT. IF WE FOLLOWED THE INITIAL TCASII; A BIT MORE AGGRESSIVELY; PERHAPS; THERE MAY NOT HAVE BEEN A LOSS OF SEPARATION (ALTHOUGH THERE MAY HAVE BEEN INJURED FLT ATTENDANTS AND PAX). THE ATC CTLR WAS QUICK TO RECOGNIZE THE SIT AND GIVE US A TURN (RATHER THAN INSISTING ON CLB). SUPPLEMENTAL INFO FROM ACN 284802: ANOTHER ACFT WAS DSNDING TOWARD US; COPLT HAD ACFT IN SIGHT AFTER THE RA. ATC CALLED ME AT HOME NEXT DAY TO ADVISE THEY HAD A LOSS OF SEPARATION POSSIBLY DUE TO PLTS FOLLOWING THEIR TCASII. I DON'T KNOW IF THIS WOULD HAVE AVOIDED THE PROB BUT I WAS A LITTLE SLOW TO RESPOND TO THE TCASII AS THE AUTOPLT WAS TRIMMED FOR A GOOD CLB AND I FOUND IT HARD TO CHANGE THE ATTITUDE OF AUTOPLT. I SHOULD HAVE PUSHED THE NOSE DOWN HARDER; BUT ALSO; I DIDN'T WANT TO BOUNCE PAX OFF THE CEILING.