FLT CREW MISUNDERSTOOD DESCENT CLRNC AND DESCENDED INTO CONFLICT WITH OPPOSITE DIRECTION TRAFFIC.

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OUR FLT WAS OPERATING AT FL370. APPROX AIR VOR THE CTLR CLRED US TO; 'CROSS 50 MI FROM EEY 270; CLRED DIRECT TO EEY; HURRY TO FL310; PLT DISCRETION AFTER THAT.' WE WERE STILL QUITE A DISTANCE FROM THAT POINT; AND A GOOD DISTANCE FROM OUR START OF DSCNT POINT. THE F/O; AT MY REQUEST; ASKED IF WE COULD JUST HAVE PLT'S DISCRETION. THE REPLY TO THE BEST OF MY KNOWLEDGE WAS; 'HURRY DOWN TO FL350 FOR NOW.' WE COMPLIED. AT FL350 WE ENCOUNTERED LIGHT TURB; BUT ALSO NOTED AN ACFT PROCEEDING LEFT TO RIGHT DIRECTLY UNDER US. AT WHAT ALT WE DIDN'T KNOW; BUT I GUESSED FL310 OR FL330. AT THIS TIME WE STILL THOUGHT WE WERE CLRED PLT'S DISCRETION TO FL270; AND I COMMENTED TO THE FO; 'I WONDER IF THE CTLR KNEW ABOUT HIM (THE OTHER ACFT)?' WE HAD BEEN LEVEL AT FL350 FOR NO MORE THAN 2 MINS; STILL ENCOUNTERING LIGHT TURB; AND AFTER THE OTHER ACFT HAD PASSED; I STARTED DSCNT SIMULTANEOUSLY ADVISING THE CTLR; 'LGT X IS PICKING UP LIGHT CHOP AND IS OUT OF 350.' NO TFC ADVISORY WAS GIVEN OF THE OTHER ACFT SHORTLY THEREAFTER WE WERE CLRED TO DESCENDED. ABOUT 5 MINS LATER WE WERE ADVISED WE HAD AN ALT EXCURSION AND IF WE DESIRED FURTHER INFO; TO CALL THE WATCH SUPVR CLE ARTCC; WHICH WE DID. THE NEXT DAY I ALSO CALLED THE QUALITY CTL OFFICER WHO EXPLAINED THE SITUATION TO ME. HIS ONLY COMMENT ABOUT THE CTLR WAS; 'HE COULD HAVE USED BETTER PHRASEOLOGY;' BUT DID SAY THE TAPES REPLAYED SAID HE TOLD US TO; 'HURRY TO FL350 NOW; LOWER LATER;' OR WORDS TO THAT EFFECT. WE DID NOT REMEMBER THAT. BOTH OF US WERE SURE HURRY TO 350 AND PLT'S DISCRETION TO 270 WAS OUR CLRNC. I WAS EQUALLY SURPRISED WHEN THE ACFT PASSED UNDER US AND THAT IT WAS AT FL330. HAD WE STARTED OUT OF FL270 AND HURRIED TO FL310 AS WAS OUR FIRST INSTRUCTION; I BELIEVE WE WOULD HAVE BEEN VERY CLOSE TO THAT ACFT. I BELIEVE THE REAL PROB IS THAT THE CTLR HAD TO HAVE US AT FL350 OR LOWER TO HAND US OFF TO THE NEXT SECTOR WHOSE BOUNDARY WAS APCHING VERY RAPIDLY; AS WE WERE STILL A LONG WAY FROM OUR TOP OF DSCNT POINT.