767 FLC CONFUSED REGARDING TAXI CLRNC TURNS ON TXWY AS OPPOSITE DIRECTION DOES THE SAME.

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AT APPROX XY05; WE CALLED GND CTL FOR PUSHBACK CLRNC FROM GATE AT LAX; AND WE WERE INSTRUCTED TO HOLD THE PUSH AND AWAIT CALLBACK. THERE WAS VERY HVY TFC ON THE RAMPS AND TXWYS. AT ABOUT XX15 WE WERE CLRED PUSHBACK ONTO TXWY K AND TO EXPECT RWY 25R FOR DEP. AFTER COMPLETING PUSHBACK AND ENG START WE CALLED FOR TAXI CLRNC AT ABOUT XX18; AND WERE INITIALLY INSTRUCTED TO WAIT FOR A COMPANY DC-10 TO PASS ON TXWY J; THEN TURN AT TXWY 42 TO TXWY J FOR TAXI TO RWY 25 R. THAT CLRNC WAS IMMEDIATELY CANCELED AND WE WERE INSTRUCTED TO HOLD POS. AT ABOUT XX25 WE RECEIVED TAXI CLRNC; WHICH I UNDERSTOOD TO BE TO TAXI OUT TXWY 42 AND TRANSITION TO THE INNER (TXWY K) AT 36. AT THE TIME THIS SEEMED REASONABLE; AS THERE WAS OPPOSITE DIRECTION TFC AHEAD OF US ON TXWY K; WHILE TXWY J WAS CLR. I INITIATED TAXI FOLLOWING THESE INSTRUCTIONS; AND AS I WAS COMPLETING MY TURN ONTO TXWY J FROM RWY 42; I SAW THAT ANOTHER ACFT (OPPOSITE DIRECTION) WAS ALSO ABOUT TO COMPLETE A TURN ONTO TXWY J AT TXWY 36. WE BOTH CAME TO A STOP UPON SEEING OUR BLOCKED PATHS. GND CTL IMMEDIATELY RECOGNIZED THE SIT AND ASKED IF EITHER OF US WAS ABLE TO REVERSE OUR DIRECTIONS TO RESOLVE THE CONFLICT. NEITHER OF US WERE IN A POS ON THE TXWYS TO SAFELY EXECUTE THE MANEUVER. GND CTL REQUESTED THAT WE COORDINATE WITH OUR COMPANY OPS TO ARRANGE FOR A TUG TO COME OUT AND PUSH US CLR OF TXWY J. THEY POINTED OUT THAT MY ORIGINAL CLRNC WAS TO TAXI FROM TXWY 42 AND TO TRANSITION FROM THE INNER TO THE OUTER AT TXWY 36. OUR TUG ARRIVED AT ABOUT XX35; AND SHORTLY THEREAFTER THE PROB WAS RESOLVED UNEVENTFULLY.