VFR GA PLT ON L BASE AT HWO FOR RWY 36R AT 400 FT EXPERIENCED NMAC WITH TFC ON R BASE FOR SAME RWY; NO TFC ISSUED BY ATC.

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WE WERE RETURNING VFR TO HWO FROM THE W. WINDS VARIABLE AT 3 KTS; VISIBILITY 10 MI; CEILING UNLIMITED; RWYS 36L AND 36R ACTIVE AT XA00. I CONTACTED THE TWR FROM 11 MI W; LNDG WITH AWOS INFO. TWR ASSIGNED ME RWY 36L; RPT L BASE. A FEW MI OUT; TWR SWITCHED ME TO RWY 36R; CLRED TO LAND; CONTINUE ON L BASE. AS I TURNED FINAL; A SMALL HIGH WING PLANE APPEARED CLOSE OFF MY R WING; AT MY ALT; ON A PARALLEL FINAL TO THE SAME RWY. I DOVE UNDER THE OTHER PLANE; RPTED 'EXCESSIVE TFC ON FINAL' AND ANNOUNCED 'TURNING R 360 DEGS.' COMPLETING MY 360 DEG TURN; TWR CLRED ME TO LAND ON RWY 36R; WHICH I DID. I DIDN'T SEE WHERE THE OTHER PLANE WENT; BUT I SUSPECT THEY NEVER SAW US AND CONTINUED LNDG ON RWY 36R. WHILE TAXIING TO MY HANGAR; I GOT A CALL FROM GND CTL. THEY APOLOGIZED FOR THE NEAR MISS AND SAID THE OTHER PLANE'S XPONDER WAS INTERMITTENT AND THEY SEEM TO HAVE LOST HIM FROM THEIR RADAR. I CAN'T EXPLAIN WHY WE DIDN'T SEE THE OTHER ACFT (NOR DID THEY SEE US) WHILE WE WERE STILL ON THE BASE LEG; AS WE MUST HAVE BEEN HEAD-ON TO EACH OTHER. IT WASN'T UNTIL WE HAD BOTH TURNED FINAL; APCHING FROM OPPOSITE DIRECTIONS (ME FROM THE W; THEY FROM THE E) THAT THE OTHER ACFT WAS OBSERVED AND EVASIVE ACTION WAS INITIATED. PERHAPS THEY HAD FLOWN A WIDER PATTERN AND WERE READY ON FINAL WHEN WE TURNED FINAL. I DON'T KNOW. WE DIDN'T SEE THEM UNTIL THE VERY END. SINCE WE WERE CLRED TO LAND; I HAD RELAXED MY NORMAL TFC SCAN BECAUSE IT WAS; AFTER ALL;; A TWR CTLED ARPT ENVIRONMENT. IN THE FUTURE; I'LL HEIGHTEN THE LEVEL OF TFC SCANNING WHEN IN THE TFC PATTERN; EVEN AT A CTLED ARPT.