B767-300 CREW AND A FOREIGN ACR HAD LESS THAN LEGAL SEPARATION IN EGTT RVSM CLASS A AIRSPACE. ARTCC RADAR CTLR MIXED UP THE CONFLICTING ACFT AND TRIED TO TURN THEM INTO EACH OTHER.

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FL350 PRIOR TO DIKAS ON UG1 TO DVR WITH LONDON ON FREQ 129.37; WAS TOLD TO MAINTAIN PRESENT HDG OF 108 DEGS. ABOUT 1 MIN LATER; ACFT X WAS TOLD TO TURN L TO HDG 100 DEGS. PLTS NOTICED AN ACFT ON TCASII AT L 9-10 O'CLOCK POS; DSNDING. WE PICKED UP ACFT Y VISUALLY AND TOLD LONDON AND ASKED THEM WHAT THEY WANTED US TO DO. LONDON INSTRUCTED US TO CONTINUE TURN TO 070 DEGS AND ACFT Y WAS TOLD TO TURN R TO 130 DEGS. WE DID NOT INITIATE OUR L TURN DUE TO TFC IN SIGHT AND ACFT Y APPEARED TO BE A CONFLICT. LONDON STOPPED ACFT Y'S DSCNT AND TOLD THEM TO CLB TO FL355. LONDON CONTINUED TO GIVE CONFLICTING TURNS TO BOTH ACFT. WE REFUSED TO TURN L INTO ACFT Y UNTIL THEY STOPPED THEIR DSCNT AND CLBED TO FL360. THEN LONDON STATED THAT HE HAD MISLABELED (SWITCHED) ACFT X AND ACFT Y ON HIS RADAR SCREEN AND ASKED US TO FILE A RPT. ACFT Y ALSO STATED THEY HAD US IN SIGHT AND DID NOT INITIATE THEIR TURN. IN RETROSPECT; THE LONDON CTLR WAS TRYING TO TURN US AWAY FROM EACH OTHER AS ACFT Y WAS DSNDING THROUGH OUR ALT; BUT IN REALITY; DUE TO CTLR ERROR; HE WAS ACTUALLY TRYING TO TURN US INTO EACH OTHER. TCASII TURNED RED -- NO AURAL ALERTS.