COMMUTER BE02 HAS FAILURE OF DIRECTIONAL GYROSCOPE; MAKES TURN IN WRONG DIRECTION.

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CREW OF 2 DEPARTED CLEVELAND HOPKINS INTL ARPT ON RWY 23 DURING VMC CONDITIONS ON AN IFR FLT PLAN; ACFT TYPE BE02 (BEECH 1900 TURBOPROP). THE TKOF; INITIAL CLB AND A TURN TO AN ASSIGNED HDG OF 180 DEGS BY CLEVELAND DEP WERE UNEVENTFUL. AT THAT TIME I (COPLT; PNF AT THE TIME) EXECUTED THE CLB CHKLIST. BOTH CREW MEMBERS EXECUTED THE CHKLIST ITEMS AS NORMAL. I THEN ATTEMPTED TO ACCOMPLISH THE STATION CALL (CALL BACK OUT AND OFF TIMES) WHICH IS PART OF THE CLB CHKLIST. I TURNED OFF THE AUDIO TO RADIO #1 ON MY SIDE IN ORDER TO MONITOR RADIO #2 AND CONFIRMED THAT THE CAPT WOULD MONITOR RADIO #1. AS I COMPLETED THE STATION CALL; THE CAPT MADE A TURN TO THE R FROM ASSIGNED HDG OF 180 DEGS AND THEN MADE AN IMMEDIATE TURN TO THE L (OPPOSITE DIRECTION). AT THIS TIME I TURNED ON RADIO #1 AND SHUT DOWN RADIO #2; AND ASKED THE CAPT WHAT WAS 'GOING ON.' HE SAID HE HAD BEEN CLRED DIRECT TO AKRON VOR; HE CENTERED THE NEEDLE WITH THE COURSE DIRECT BUTTON. IT SHOWED A TURN TO THE R AND HE TOOK A HDG IN THAT DIRECTION. HE STATED ATC TOLD HIM HE TURNED THE WRONG DIRECTION AND THAT AKRON VOR WAS SW OF THE ARPT WHICH WOULD BE A L TURN (FACT). AT THAT TIME I REALIZED OUR DIRECTIONAL GYROSCOPES WERE DIFFERENT AND THAT THE CAPT'S DIRECTIONAL GYROSCOPE HAD A HDG FLAG. THE HDG DIFFERENCE WAS IN EXCESS OF 90 DEGS. THE CAPT STATED THAT AFTER HE WAS CLRED DIRECT TO AKRON VOR HE SIMPLY PUSHED THE COURSE DIRECTOR BUTTON AND TURNED TOWARD THE HEAD OF THE NEEDLE. THE DIRECTIONAL GYROSCOPE ON HIS SIDE WAS IN ERROR AND HE THEREFORE TURNED THE WRONG DIRECTION. HAD I BEEN MONITORING RADIO #1 AT THIS TIME THIS SIT MAY HAVE BEEN AVOIDED. HOWEVER; THE STATION CALL IS AN ASSIGNED DUTY OF THE PNF DURING THIS PHASE OF FLT. PERHAPS THE STATION CALL SHOULD NOT BE EXECUTED DURING THIS STAGE OF FLT?