RKRR ARTCC RADAR LOST SEPARATION BTWN DSNDING AND CLBING ACR'S.

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AT APPROX XA53Z; WE WERE ENRTE FROM HKG TO ICN. NAVING NBOUND ON AIRWAY B576 BTWN CJU AND KWA VORS; INCHEON CTL (RKRR) DIRECTED US TO DSND FROM FL330 TO FL290. WHILE DSNDING THROUGH FL310; AND BTWN 40-36 DME S OF KWA; AN ACFT APPEARED ON OUR TCASII DISPLAY AT THE 12 - 1 O'CLOCK POS; INDICATING -900 FT AND REDUCING VERT SEPARATION RAPIDLY. BECAUSE THE WX WAS VMC; OUR CREW SAW THE OPPOSING ACFT. WE LEVELED OFF; BUT THE OPPOSING ACFT APPEARED TO BE CLBING INTO OUR FLT PATH. WE THEN IMMEDIATELY BEGAN A L TURN AWAY FROM THE TFC AND OBSERVED THE OPPOSING TFC PERFORMING THE SAME MANEUVER (TURNING L AWAY FROM US). DURING THIS TIME; THE TCASII TURNED RED IN COLOR AND ANNOUNCED TFC; BUT DID NOT GIVE A CLB OR DSND COMMAND. AT THE SAME TIME; INCHEON CTL DIRECTED US TO TURN R; WHICH WOULD HAVE TURNED US INTO THE TFC. WE CONTINUED TO DISREGARD INCHEON CTL'S INSTRUCTIONS AND TOOK OUR OWN EVASIVE ACTION TO AVOID A COLLISION. IT IS OUR ESTIMATE THAT HORIZ SEPARATION WAS NO CLOSER THAN 2000 FT WHEN THE ACFT PASSED EACH OTHER. THE TCASII ANNOUNCED 'CLR OF TFC' AFTER WE RESOLVED THE CONFLICT. AFTER RETURNING TO COURSE AND CONTINUING DSCNT TO FL290; WE QUERIED INCHEON CTL AND VOICED A COMPLAINT. INCHEON CTL TEMPORARILY DISMISSED HIMSELF; THEN RPTED TO US THAT HE HAD MADE A MISTAKE AND APOLOGIZED. INCHEON CTL ASKED IF WE HAD RECEIVED A TCASII RA AND WE ANSWERED IN THE AFFIRMATIVE. HE APOLOGIZED AGAIN (IT SOUNDED LIKE THE SAME CTLR). UPON ARRIVING IN ICN; I CALLED GCC; SPOKE WITH THE SUPVR AND GAVE NOTICE THAT I WOULD BE FILING THIS RPT.