ZNY ARTCC RADAR CTLR INSTRUCTS A DC10 TO MAKE SEVERAL TURNS TO AVOID A CONFLICT WITH A CITATION OPPOSITE DIRECTION. BOTH ACFT AT FL310.

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ENRTE FROM EWR TO IND ON J80; ZNY CLRED US DIRECT MIE; THEN CLB TO FL310. PASSING FL305; CTR ISSUED 'TURN R 30 DEGS; OPPOSITE DIRECTION TFC' QUICKLY FOLLOWED BY 'TURN L 30 DEGS.' TCASII DISPLAYED A TARGET AT 12 O'CLOCK POS; 5 MI; +400 FT EBOUND. THE AUTOPLT WAS TURNING AND INITIATING THE LEVELOFF WHEN THE TCASII ISSUED A CLB RA WITH THE TFC JUST ABOVE US. A DIFFERENT VOICE FROM ZNY RECLRED US 'TURN R 30 DEGS AND CONTACT ZOB.' THE TCASII REVERSED ITS COMMAND WITH A DSCNT RA. THE TFC WAS LESS THAN 2 MI AND +100 FT. THE CAPT DISCONNECTED THE AUTOPLT AND STARTED A GENTLE DSCNT. THE SO AND I WERE TRYING TO GET A VISUAL ON THE TFC. HE SIGHTED IT OFF THE L SIDE AND IDENTED IT AS A CESSNA CITATION. HE ESTIMATED IT PASSED WITHIN 1/2 MI AND +100 FT ABOVE US. TCASII CONFIRMED THIS. WE DID NOT HEAR ANY XMISSIONS DIRECTED AT THE CITATION AND BELIEVE HE WAS ON A DIFFERENT FREQ. HE WAS ALSO AT THE WRONG ALT FOR DIRECTION OF FLT. I BELIEVE CTLR TRAINING WAS IN PROGRESS AND AN INCORRECT XFER OF COM OCCURRED.