2 ACR MD80'S ALMOST COLLIDE ON RWY 24L AND TXWY Z IN LAX.

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RWY INCURSION. CHKED IN WITH LAX TWR JUST PRIOR TO ROMEM LOM FOR RWY 24R WITH NO RESPONSE. CHKED IN AFTER ROMEN WITH NO RESPONSE. LAX TWR CLRED US TO LAND AT JUST BELOW 1000 FT. DURING LNDG ROLLOUT; TWR CLRED US TO CROSS RWY 24L. WE REQUESTED REVERSE HIGH SPD WITH NO RESPONSE FROM TWR. TWR AGAIN CLRED US ACROSS RWY 24L CONTACT GND ON 121.65. AS WE ENTERED RWY 24L WE HEARD TWR CALL FOR ACR ABC TO ABORT TKOF ON RWY 24L. WE WERE UNABLE TO SEE DEP END OF RWY 24L. WE OVERHEARD ACR ABC ACKNOWLEDGE ABORT AND WE CLRED RWY 24L EXPEDITIOUSLY. AFTER GATE ARR; CAPT CALLED LAX TWR. LAX TWR SUPVR STATED POTENTIAL OPERROR ON LAX TWR'S PART. SUPPLEMENTAL INFO FROM ACN 492724: WHILE ON LNDG ROLLOUT ON RWY 24R WE WERE CLRED TO CROSS RWY 24L AND CONTACT GND OTHER SIDE. WE REQUESTED TO USE THE REVERSE HIGH SPD; AND RECEIVED NO REPLY. WE PROCEEDED TOWARDS THE FORWARD HIGH SPD AND BEFORE REACHING THE HOLD SHORT RWY 24L BAR; WERE AGAIN CLRED TO CROSS RWY 24L AND CONTACT GND 121.65 THE OTHER SIDE. THE FO ACKNOWLEDGED WITH A READBACK AND THE TAXI CONTINUED. APPROX 1/3 ONTO RWY 24L WE HEARD THE CTLR ANNOUNCING AN ABORT FOR AN ACFT ON RWY 24L. BY THIS TIME WE WERE APPROX 1/2 TO 2/3 ACROSS THE RWY 24L AND EXPEDITED THE REST OF THE WAY ACROSS. DUE TO THE ANGLE OF THE HIGH SPD AND THE NIGHT CONDITIONS IT WAS NOT POSSIBLE TO SEE THE ACFT AT THE DEP END OF RWY 24L PRIOR TO THIS EVENT. THE TWR WAS CONTACTED AFTER SHUTDOWN. THEY HAD PULLED THE TAPES AND ADVISED THE CAPT THAT 'THEY WERE INVESTIGATING AN OPERROR ON THEIR PART.' SUPPLEMENTAL INFO FROM ACN 492720: WE WERE CLRED FOR TKOF ON RWY 24L. AT 90 KTS TWR CANCELED CLRNC DUE TO ANOTHER ACR XING DOWNFIELD. WE ABORTED AND RETURNED TO GATE FOR BRAKE INSPECTION. THE CAPT OF THE XING ACFT CAME BY THE GATE AND TOLD US HE HAD BEEN CLRED BY ANOTHER CTLR TO CROSS RWY 24L AFTER HIS LNDG ROLLOUT ON RWY 24R. AFTER TALKING TO TWR; HE SAID THAT TWR ACKNOWLEDGED IT WAS THEIR OPERROR.