A TURNING THE WRONG DIRECTION BY THE PIC OF A B737 AT FL330 PLACES HIS ACFT IN POTENTIAL CONFLICT WITH ANOTHER ACR FLT AT THE SAME ALT; 80 NM FROM MEM; TN.

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DURING CRUISE AT FL300; CTR REQUESTED A TURN 30 DEGS AT THE SAME TIME THE FO WAS HAVING A COUGHING FIT. IT WAS THE FO WORKING THE RADIOS AS I WAS FLYING THE LEG. I LOOKED AT THE FO AND WANTED TO SEE IF HE HAD HEARD THE REQUEST AND IF HE WAS GOING TO RESPOND. HE WAS STILL COUGHING; SO; AS I WAS TOLD BY CTR; I RESPONDED '30 DEGS L; FLT X;' BUT MADE A 30 DEG R TURN INSTEAD. DURING THE NEXT 30 SECONDS OR SO; CTR WAS GIVING TURN REQUESTS TO ANOTHER AIRLINE'S FLT BUT AIRLINE FLT Y WAS NOT RESPONDING. I REMEMBER CTR SAYING 'FLT Y; TURN 30 DEGS;' BUT I DO NOT REMEMBER THE FLT NUMBER OR DIRECTION OF TURN REQUESTED BY ZME. THERE WERE 3 ATTEMPTS TO CONTACT FLT Y AND I HEARD NO RESPONSE. JUST AFTER THE THIRD ATTEMPT TO CONTACT FLT Y; I RECEIVED INSTRUCTIONS FROM ZME TO 'TURN L IMMEDIATELY' AND 'DESCEND TO FL310;' WHICH I DID. I TURNED ABOUT 60 DEGS L AND WHEN BEGINNING THE DSCNT RECEIVED A TA. AFTER A MIN OR SO; I CLBED BACK TO FL330 AND PROCEEDED ON COURSE AS REQUESTED BY CTR. I NEVER SAW FLT Y. THIS WAS A UNIQUE COMBINATION OF ERRORS. MY FO WAS COUGHING AND SNEEZING AT AN ALARMING RATE AND WAS NOT VERY HELPFUL WITH CONFIRMING RADIO CALLS OR INSTRUCTIONS. HE WENT HOME SICK THE NEXT DAY. THE CTLR DID NOT GIVE REASON FOR THE HDG CHANGE. NUMEROUS ACFT HAD BEEN REQUESTING HDG CHANGES FOR WX. I USUALLY RECEIVE HDG CHANGES FOR 'IN TRAIL SPACING;' AND DID NOT KNOW THIS REQUEST WAS FOR CONVERGING TFC AT THE SAME ALT. THE CTR DID NOT FOLLOW MY COURSE CHANGE AND QUESTIONED THE DIRECTION OF MY TURN. IT IS MUCH SAFER TO ASSIGN A MAGNETIC HDG AFTER ASKING THE ACFT THE PRESENT HDG; WHICH IS ROUTINE. FLT Y WAS EITHER NOT ON THE FREQ AND SHOULD HAVE BEEN OR THE CTLR WAS WORKING NUMEROUS FREQS. SINCE THE OTHER ACFT WAS IN CLOSE PROX TO MY ACFT AND AT THE SAME ALT; I WOULD EXPECT THE ACFT TO BE SHARING THE SAME FREQ I WAS SHARING WITH THE CTLR.