AN ACR B737 FLC BACK TAXIED ON THE WRONG RWY AND HAD A CLOSE ENCOUNTER WITH ANOTHER ACR'S B737 THAT WAS ON ITS TKOF RUN.

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I WAS THE CAPT ON ACR X. WE PUSHED BACK FROM THE GATE ONTO TXWY N; FACING EBOUND W OF TXWYS F AND K. THE FO CALLED ONTARIO GND CTL FOR TAXI. ONTARIO GND GAVE HIM THE TAXI INSTRUCTIONS; HE READ THEM BACK; AND I REPEATED THEM TO HIM. AFTER THE FACT; I BELIEVE THE FO READ BACK THE PROPER INSTRUCTION TO BACK TAXI ON RWY 26R AND I REPEATED THE PROPER INSTRUCTION TO HIM. UNINTENTIONALLY; I TRANSPOSED THE RWYS AND HAD THE PICTURE I WAS TO BACK TAXI ON RWY 26L; THE S RWY. WE PROCEEDED ONTO TXWY K; THE FO PUT IN TWR FREQ AS WE ENTERED RWY 26R; AND I CONTINUED ON TXWY K AND MADE A L TURN DOWN RWY 26L. WE DID NOT HEAR ACR Y GET CLRED FOR TKOF. THE RPTED VISIBILITY WAS 1/4 MI IN FOG. WE WERE NOT PERFORMING ANY DUTIES AT THAT POINT AND MY WHOLE FOCUS WAS ON VISUALLY CLRING THE RWY. I NOTICED TWO DIM LIGHTS AT THE E END; FOLLOWED QUICKLY BY AN ACFT SHAPE. THE ACFT APPEARED TO BE AT THE END AND STATIONARY. I WENT FOR TXWY P TO GET OFF THE RWY. AS I STARTED THE R TURN TO GET OFF AT TXWY P; I COULD TELL THE ACFT HAD ROLLED. HALFWAY THROUGH MY TURN; THE ACFT WAS STILL AT LEAST 3000 FT AWAY AND I KNEW I COULD BE OUT OF THE WAY. IN MY TURN; ACR Y MADE THE 'ABORTING' RADIO CALL. I CLRED AT TXWY P AND TOLD TWR I SAW ACR Y AND WAS CLR OF THE RWY. ON TXWY P I WATCHED ACR Y FINISH THE ABORT; AND THEN CLR THE RWY E OF ME ON TXWY Q. WE DID NOT CROSS PATHS AND HAD APPROX 2000 FT SEPARATION OF UNUSED RWY BTWN US. ACR Y DID A GOOD JOB. WHILE ACR Y WAS STILL ON THE RWY; AN ACR Z WHO HAD BEEN INSTRUCTED TO FOLLOW US FOR TAXI TRIED TWICE TO CONFIRM HIS TAXI INSTRUCTIONS WITH TWR. THE FOG BEGAN CLRING RAPIDLY AND I COULD SEE HIM ON RWY 26R TAXIING EBOUND. I SWITCHED TO GND CTL AND TOLD THEM I WAS CLR OF THE RWY AND MAY HAVE MISUNDERSTOOD MY TAXI INSTRUCTIONS. I WAS TOLD TO HOLD MY POS AND I HELD ON TXWY S ABEAM TXWY P FACING E AND WATCHED ACR Y CLR THE RWY. EVENTUALLY; I FOLLOWED ACR Z TO RWY 26L. HE TAXIED BACK TO THE GATE FOR BRAKE COOLING AND I DEPARTED ONTARIO. THE PROB WAS CAUSED BY MY UNINTENTIONAL MISTAKE WHEN WE RECEIVED THE TAXI INSTRUCTION; THINKING IT WAS FOR THE S RWY. THE PATCHY FOG AT TXWY K AND RWY 26R CONFUSED THE FO WHO THOUGHT WE DID TURN DOWN RWY 26R. A POSSIBLE SOLUTION WOULD BE TO NOT USE RWYS TO BACK TAXI IN FOG SITS WHEN A TXWY IS AVAILABLE. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THIS RPTR IS A B737-300 CAPT FOR A MAJOR ACR. HE ADMITS TO BECOMING CONFUSED ABOUT THE ARPT LAYOUT AND HE BACK TAXIED ON THE WRONG RWY. HE DID SEE THE LIGHTS OF ACR Y AT THE END OF THE RWY AND AT THAT TIME HE SAID HE KNEW THAT SOMETHING WAS WRONG AND HE BEGAN THE TURN OFF OF THE RWY. HE SAID THAT HE WAS CLR OF THE RWY AS THE 'ABORT' CALL WAS MADE BY ACR Y. THIS CAPT SAID THAT HE MADE A SERIOUS MISTAKE; BUT HE THINKS THAT THE EVENT WAS PROPERLY HANDLED BY HIM ONCE THE ERROR WAS RECOGNIZED. HE WAS TOLD BY AN FAA INVESTIGATOR THAT ONT HAD REQUESTED A GND SURVEILLANCE RADAR TO BE INSTALLED THE DAY AFTER THIS EVENT. SUPPLEMENTAL INFO FROM ACN 393367: AS THE AIRPLANE ENTERED ONTO RWY 26R I TURNED TO CHANGE THE RADIO FREQ TO TWR AS INSTRUCTED. AFTER I CHANGED THE RADIO FREQ I TURNED TO MY R TO COMPLETE OUR TKOF PERFORMANCE DATA UTILIZING THE ONBOARD PERFORMANCE COMPUTER. ONCE COMPLETED I PLACED THE COMPUTER FOR THE CAPT'S VIEWING. WHEN I LOOKED OUT THE CAPT HAD DECIDED TO CLR THE RWY AT TXWY P REALIZING THE MISTAKE. SUPPLEMENTAL INFO FROM ACN 393366: WE WERE CLRED TO TAXI TO RWY 26L VIA BACK TAXI RWY 26R. PATCHES OF FOG ON THE RAMP AREA AND THE W END OF THE ARPT REDUCED VISIBILITY TO 1500-2000 FT. WE WERE CLRED FOR TKOF ON RWY 26L. AS WE WERE ROLLING DOWN THE RWY APCHING 125 KIAS; AN ACR X B737 APPEARED OUT OF THE FOG BANK TAXIING DIRECTLY TOWARDS US ON THE RWY APPROX 3500 FT AHEAD. WE REJECTED THE TKOF WITH MAX BRAKING AND CLRED THE RWY TO THE S ON TXWY Q MISSING THE ACR X JET BY APPROX 300 FT. ACR X CLRED SIMULTANEOUSLY WITH US TO THE S ALSO ON TXWY P. BOTH ACFT CAME TO A STOP WINGTIP TO WINGTIP APPROX 2 ACFT WIDTHS APART. ACR X'S CAPT HAD EVIDENTLY GOTTEN LOST AND CROSSED RWY 26R ON TXWY K AND HAD TAXIED ONTO RWY 26L THINKING HE WAS ON RWY 26R. WHAT MADE THE SIT WORSE WAS HE HAD NO EXTERIOR LIGHTS ON EXCEPT HIS RED ANTI-COLLISION LIGHT; NOT EVEN HIS TAXI LIGHT. REMEMBER HE WAS TAXIING IN THE FOG WITH VISIBILITY 1500-2000 FT. CALLBACK CONVERSATION WITH RPTR OF ACN 393366 REVEALED THE FOLLOWING INFO: THE RPTR IS A B737- 300 CAPT WHO CLAIMS TO HAVE HAD SEVERAL ENCOUNTERS WITH ANOTHER ACR'S PLTS (ACR X). IN THIS CASE; THE REJECTED TKOF WAS SUCCESSFUL; BUT THE RPTR IS CONCERNED ABOUT THE ACR X'S PLT'S ATTITUDE AFTER THE EVENT AND THE GENERAL ACR X OPERATIONAL PHILOSOPHY. THE CAPT SAID THAT HE HAD TO BRAKE THE ACFT ABRUPTLY TO AVOID A COLLISION AND AFTER TURNING OFF OF THE RWY HE DECIDED TO RETURN TO THE GATE TO WAIT OUT THE BRAKE COOLING PERIOD. SUPPLEMENTAL INFO FROM ACN 393229: WE BEGAN OUR TKOF ROLL ON RWY 26L WITH ABOUT 5000 FT OF VISIBILITY WITH FOG AT THE DEP END OF THE RWY. AT APPROX 120 KIAS; AN ACR X B737; APPEARED APPROX 4000 FT IN FRONT OF US ON RWY 26L COMING HEAD-ON ON CTRLINE. CAPT IMMEDIATELY ABORTED. WE CAME WITHIN APPROX 500-800 FT OF ACR X WHEN WE CLRED THE RWY TO THE S. SUPPLEMENTAL INFO FROM ACN 393511: VISIBILITY VARIED FROM GOOD TO VERY POOR WITH RVR RPTS VARYING UP AND DOWN FROM 800 FT TO 5000+ FT. WE PUSHED BACK THE SAME TIME AS AN ACR X B737. ACR X BEGAN TAXIING AND OUR TAXI CLRNC WAS 'FOLLOW ACR X; TXWY K; BACK TAXI ON RWY 26R FOR RWY 26L; HOLD EBOUND; MONITOR TWR ON 120.6.' WE DO NOT RECALL ACR X'S TAXI INSTRUCTIONS. VISIBILITY HAD GONE DOWN; AND AS SOON AS ACR X BEGAN TO TAXI; HE DISAPPEARED. WE TURNED ONTO RWY 26R TO BACK TAXI; AND A RAPID SEQUENCE OF EVENTS OCCURRED WHEN WE SWITCHED TO MONITOR TWR ON 120.6. AN ACR Y B737 WAS CLRED FOR TKOF ON RWY 26L. WE WERE UNABLE TO SEE THE ACR X B737 WE WERE TO FOLLOW. AT THAT POINT WE OBSERVED AN ACR X B737 TAXIING ON WHAT APPEARED TO BE RWY 26L; JUST AHEAD AND TO OUR R. AT THE SAME TIME ACR Y DECLARED THEY WERE ABORTING (HIGH SPD) WITH ANOTHER ACFT ON THE RWY. SIMULTANEOUSLY; THE ACR X JET MADE A RAPID TURN OFF THE RWY AT TXWY P OR Q; IT WAS DIFFICULT TO TELL. AS SOON AS WE COULD; WE RECONFIRMED OUR BACK TAXI CLRNC ON RWY 26R. THIS WAS A CLOSE CALL; AND COULD EASILY BEEN CATASTROPHIC. EVEN THOUGH WE; ACR Z WERE ON THE PARALLEL RWY; WE COULD EASILY HAVE BEEN INVOLVED; AS THE COLLISION WOULD HAVE OCCURRED JUST AHEAD AND ABEAM US. CONTRIBUTING FACTORS: ARPT CONSTRUCTION; NO GND RADAR; TWR UNABLE TO SEE ACFT INVOLVED; MISUNDERSTOOD CLRNC.