SMA ACFT EXPERIENCED CONFLICT AT 3500 FT DURING OPS MISSION WHEN BOTH CLE TRACON AND TWR REPORTEDLY ISSUED CLRNCS THAT CAUSED THE CONFLICT.

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ACFT AND CREW WERE ON A MISSION. PER AN ESTABLISHED LOA WITH THE LCL TRACON; THE CREW CONTACTED TRACON PRIOR TO THE FLT. TRACON WAS ADVISED THAT WE WERE WORKING AT 4500 FT MSL 2 MI NW OF CLE. TRACON; PER OUR PROCS; GAVE US A CLRNC OVER THE PHONE; CLRING US TO 4500 FT; WITHIN THE CLASS B AND ADVISED WE STAY WITH TWR ON 120.9. AFTER TKOF; TWR TOLD US TO MAINTAIN RWY HDG (240 DEGS); THEN CONTACT APCH. WE STAYED ON ASSIGNED HDG; CONTINUED OUR CLB; THEN CALLED APCH. UPON CONTACT; APCH GAVE US 280 DEG HDG AND WE CONTINUED CLBING WHILE WE TURNED. AFTER SOME TIME; 3-4 MI MAYBE; WE ASKED APCH IF WE COULD TURN TO GO ON STATION (2 MI NW OF ZZZ); APCH REPLIED STAND BY. HE TOLD US TO TURN TOWARD THE FIELD AND CONTACT TWR. AFTER WE TURNED TOWARD THE FIELD; TWR WAS CALLED. ASKED US TO LEVEL OFF AT 3500 FT WITH A TURN TO 180 DEGS. WE DSNDED OUT OF 3800 FT TO 3500 FT DURING THE TURN. OUR TIS SYS WAS ALARMING; THEN WE RECEIVED A HDG OF 360 DEGS. AS WE TURNED; WE STOPPED AT 090 DEGS AS TFC PASSED ONCOMING ON THE L SIDE OF THE ACFT AT OUR ALT AND WITHIN 500 FT. IF WE HAD COMPLETED THE TURN; WE WOULD HAVE HIT THE ONCOMING TFC. 1) TRACON DID NOT PASS OUR FLT INTENTIONS ON TO THE APPROPRIATE PEOPLE; WHICH LED TO CONFUSION FROM THE START. 2) DEP GAVE US A TURN TOWARD THE FIELD INTO THE AIRPLANE DEPS; BUT IT WAS TWR THAT DSNDED US INTO THE TFC. IF WE WOULD HAVE CONTINUED THE CLB TO 4500 FT; THERE WOULD NOT HAVE BEEN A NEAR MISS.