DURING AN ENRTE ALT CHANGE; A POTENTIAL CONFLICT WITH REDUCED CLRNC OCCURS REQUIRING A DSCNT AND TURN BY FLT X AT OR NEAR FL320; OPPOSITE DIRECTION TFC CONFLICT 115 NM W OF ELP; TX.

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DURING CRUISE AT FL370; WE WERE GIVEN A DSCNT TO A LOWER ALT WHICH WE THOUGHT WAS FL310 AND WHICH WE HAD SET AND CONFIRMED IN THE ALT WINDOW. APPARENTLY; OUR ASSIGNED ALT WAS FL330. I WAS COMMUNICATING WITH FIRST FLT ATTENDANT ON THE INTERCOM WHEN THE CAPT INITIATED A CLB AS ADVISED BY THE CTLR. NEXT MOMENT WE RECEIVED AN RA TO DSND. THE CAPT COMPLIED WITH THE RA. A FEW MOMENTS LATER; THE CTLR ADVISED US TO TURN R SHARPLY. AT THE SAME TIME WE MADE VISUAL CONTACT WITH THE TFC AND WERE ABLE TO MAINTAIN SEPARATION. I BELIEVE THE OPPOSING INSTRUCTIONS FROM THE CTLR AND TCASII FURTHER AGGRAVATED THE SIT. AN INITIAL INSTRUCTION TO TURN WOULD HAVE AVOIDED THE CTLR/TCASII CONFLICT. I AM; WITH THIS; NOT PUTTING ANY BLAME ON THE CTLR. SUPPLEMENTAL INFO FROM ACN 575368: ABOUT 150 MI W OF ELP; CTR GAVE US A CLRNC TO DSND TO WHAT I BELIEVED TO BE FL310. I DIALED FL310 INTO THE ALT ALERTER AND VERIFIED WITH THE PNF. HE AGREED. PASSING AROUND FL318; THE CTLR ASKED OUR ASSIGNED ALT. I SAID FL310. HE SAID TO CLB TO FL330. I INITIATED THE CLB; DISCONNECTED THE AUTOPLT; AND CLBED AT A RATE OF OVER 2000 FPM. AFTER SEVERAL SECONDS; THE TCASII STATED; 'TFC; DSND.' I IMMEDIATELY STOPPED DSCNT AND SAW THE OTHER ACFT PASS OFF OUR R ABOUT 1 1/2 MI AND 500 FT. I CALLED THE CTLR ON THE GND AND HE SAID HE LISTENED TO THE TAPES AND OUR READBACK WAS ACTUALLY FL330. AND WE WERE GIVEN A FREQ CHANGE; WHICH WE NEVER CHKED IN ON. THE FLT ATTENDANT HAD JUST CALLED AS WE STARTED THE DSCNT AND THE PNF ANSWERED HER ON THE INTERPHONE. I FEEL WE PROCEDURALLY WERE CORRECT. I VERIFIED WITH THE FO THE ALT; HOWEVER; HE EITHER WASN'T PAYING ATTN OR ACTUALLY THOUGHT WE WERE CLRED TO FL310.