ACR X (TCASII RA DSND) HAD POTENTIAL CONFLICT WITH OPPOSITE DIRECTION TFC SAME ALT ACR Y TCASII RA CLB. EVASIVE ACTION TAKEN.

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IN CRUISE FLT AT FL330 INBOUND TO FLM VOR ON A FLT FROM CHARLOTTE TO DAYTON. TCASII FIRST CALLED 'TFC.' TARGET WAS DISPLAYED ON TCASII SCOPE AT ABOUT 11:30 POS AT SAME ALT AND LEVEL. ATC THEN ISSUED A DSCNT FROM FL330 TO FL240 FOR US (ACR X) AND ISSUED A L TURN TO THE OTHER ACFT (ACR Y) TO A HDG OF 360 DEG. THE OTHER ACFT APPEARED TO BE ON AN EBOUND HDG PRIOR TO THE L TURN. TCASII THEN ISSUED US A RA TO DSND AT 3000 FPM. WE HAD ALREADY STARTED OUR DSCNT AND INCREASED THE RATE TO COMPLY WITH THE TCASII RA. WE HAD THE OTHER ACFT IN SIGHT AND OBSERVED HIM IN A L TURN. HE ADVISED ATC THAT HE HAD A 'TCASII ALERT' AND APPARENTLY CLBED BECAUSE SHORTLY AFTERWARD; ATC CLRED HIM TO DSND BACK TO FL330. ESTIMATED HORIZ SEP WAS 5 NM ON THE TCASII SCOPE AND WE WERE THROUGH 31000 AT THAT TIME IN OUR DSCNT. THE ENTIRE EVENT HAPPENED IN A MATTER OF SECONDS; PROBABLY LESS THAN 20 SECONDS; FROM 'TFC; TFC' ALERT ON THE TCASII UNTIL 'CLR OF CONFLICT' ON TCASII. SUPPLEMENTAL INFO FROM ACN 213013. UPON LNDG; I CALLED OUR DISPATCH TO TALK TO ZID SUPVR. THEN TALKED TO SUPVR AT ZID WHO SAID 'THEY WERE LISTENING TO THE TAPES.' AND 'IT APPEARED IT WAS A FAA CTLR PROBLEM.' I CLARIFIED THAT WE HAD NOT HEARD ANY CLRNC ASSIGNED TO US UNTIL THE 'IMMEDIATE DSCNT TO FL240.' THE PROBLEM APPEARS TO BE ATC HAD US AT FL330 ON A 'WESTERLY FLT' AND ACR Y WAS AT FL330 ON AN 'EASTERLY FLT.'