MLG ACR CREW WAS ISSUED AN ALT AND A HDG BY THE TRACON DEP CTLR. AS CREW WAS TURNING TO THE HDG; THE CTLR QUESTIONED WHERE THEY WERE GOING.

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DEPARTED RWY 5 AT CLE. INITIAL HDG OF 095 DEG VECTOR FROM TWR. DURING CLB; CTLR ON DEP CTL GAVE A CLB TO 12000 FT AND R TURN TO 280 DEGS; WHICH WE ACKNOWLEDGED AND BEGAN TO EXECUTE. DURING THE TURN; THE CTLR ASKED ABOUT OUR HDG. WE REPLIED 'IN THE TURN TO 280 DEGS.' WE WERE TURNING THROUGH 210 DEGS AND CTLR TOLD US TO TURN L TO 140 DEGS AND LEVEL OFF AT 6000 FT. WE COMPLIED. NOT SURE WHY CTLR STOPPED OUR TURN SO WE ASKED IF EVERYTHING WAS OK. HE SAID HE DIDN'T THINK HE HAD GIVEN US A TURN TO 280 DEGS. WE SAID WE THOUGHT THAT HE HAD. SUPPLEMENTAL INFO FROM ACN 419063: DEP GAVE US A CLB TO 12000 FT AND A TURN TO THE R OF 280 DEGS. I READ BACK THE ALT AND THE HDG (12000 FT AND 280 DEGS). HALF WAY THROUGH THE TURN; THE CTLR LEVELED US AT 6000 FT AND GAVE US A HDG OF 140 DEGS. HE ASKED IF WE WERE TURNING TO 240 DEGS (HAPPENED TO BE THE HDG TO MANSFIELD; OUR FIRST FIX). I SAID NO; WE WERE GIVEN 280 DEGS AND WERE TURNING TOWARDS THAT. THE CTLR REPLIED THAT HE DIDN'T REMEMBER GIVING US THAT HDG. WE REMEMBER HIM GIVING THE HDG OF 280 DEGS AND WE REMEMBER READING IT BACK.