POTENTIAL CONFLICT BTWN 2 ACR ACFT ENSUES ON ZNY FREQ DURING AN ALTDEV BY ONE AND A WRONG DIRECTION TURN BY THE OTHER WHILE 100 NM W OF HAYED INTXN; PA.

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I CALLED ATC TO ASK FOR LOWER ALT [WE WERE AT FL350] TO CROSS HAYED AT FL180. ATC THEN CLRED ME DIRECT TO SWEET INTXN AND CROSS AT 9000 FT. ABOUT 1 MIN LATER; ATC SAID 'MAKE A HARD TURN TO THE L 050 DEGS.' WE THOUGHT HE SAID TURN R 050 DEGS. I THOUGHT IT WAS STRANGE; BUT STARTED THE TURN. ATC WAS VERY RATTLED AND; IT SEEMED; SPOKE VERY FAST. (WE READ BACK TURN R ALSO.) WHEN ATC CLRED US DIRECT TO SWEET; HE SET UP A TFC CONFLICT OF HIS OWN MAKING AND THEN GOT VERY EXCITED ABOUT GETTING IT FIXED. ANYWAY; BY TURNING R INSTEAD OF L WE CAME WITHIN 5 MI OF ANOTHER ACFT. ATC ALSO SAID THEY CLRED US TO 330 DURING THE TURN BUT NEITHER I NOR MY COPLT HEARD THAT PART. SUPPLEMENTAL INFO FROM ACN 539671: ACR Y; BELOW AND BEHIND US; CLBING; WAS CLRED ONLY TO FL330 BUT WAS CLBING TO FL370. BELLS AND WHISTLES SOUNDED IN ZNY FOR THE TFC CONFLICT WITH OUR ACFT. NEVER SAW OR HAD TCASII ALERT WITH ACFT. CTLR WORKING OUR FLT TOLD US TO MAKE A TIGHT R TURN TO 170 DEGS. IN THE TURN; ASKED US IF WE WERE IN THE TIGHT L TURN TO 070 DEGS. I SAID 'NO; YOU TOLD US TURN R.' HE SAID 'TURN IMMEDIATELY L TO 070 DEGS AND DSND TO FL330.' CTLR WAS NOW LOSING IT WHEN ANOTHER CTLR CAME ON AND GAVE US A HDG AND ALT TO FL290. NEVER HAD A TCASII WARNING OR RA.